arxiv:1707.09368v2 [math.rt] 11 oct 2017 · pdf filemoscovici, gromov, higson and ... and...

arX

iv:1

707.

0936

8v2

[m

ath.

RT

] 1

1 O

ct 2

017

COMMENTS ON MY PAPERS

G. Lusztig

Abstract. Changes have been made to the comments on [37],[40],[54],[57],[95],[110].

Comments to [49] have been added.

This document contain comments on some of my papers. I hope to add to itmore comments in future versions.

[8] (with J.Milnor and F.P.Peterson)Semicharacteristics and cobordism, 1969

I did the work on this paper during a two months stay in Oxford (fall of1968). During my first meeting with Atiyah, he and Singer explained to methe following question on the (Kervaire) semicharacteristic. A compact smoothoriented manifold M of dimension 4n + 1 has a semicharacteristic c(M, p) =∑

i∈[0,2n] dimHi(M, k) mod 2 with respect to a field k of characteristic p ≥ 0.

At the time it was known that the obstruction to the existence of two independentvector fields on M is equal to c(M, 2) if M is spin [E. Thomas, Bull. Amer. Math.Soc. 1969] and to c(M, 0), without assumption (Atiyah-Singer). The question waswhether there is a relation between c(M, 0) and c(M, 2) which would make clearthat the Atiyah-Singer result implies the Thomas result. (A few months earlierthey asked F.Peterson at MIT the same question.) Then I and (independently)F.Peterson and J.Milnor found (different) formulas for c(M, 0) − c(M, 2); one ofthose formulas expressed c(M, 0)−c(M, 2) as the Stiefel-Whitney number w2w4n−1

which clearly vanishes for spin manifolds. The initial proofs of both formulas usedWall’s results on the structure of the oriented cobordism group (that is the formu-las were checked on the generators of that group) but in the final version Wall’sresults are not used. The result of this paper is used in [Atiyah,Singer, Ann.Math.1971] and is generalized in [10].

Supported in part by NSF grant DMS-1566618

Typeset by AMS-TEX

1

http://arxiv.org/abs/1707.09368v2

2 G. LUSZTIG

[9] Remarks on the holomorphic Lefschetz formula, 1969

I did the work on this paper during a two months stay in Oxford (fall of 1968).The main result of the paper is the following rigidity result: the holomorphicLefschetz number of a circle action on a compact complex manifold is constant.(This is of interest only in the case where the manifold is non-Kaehler, when theinduced action on the Dolbeault cohomology may be nonconstant.) The argumentthat I used in this paper has played a role in the proof of the vanishing of theA-genus of a 4k-dimensional spin manifold with nontrivial circle action given in[Atiyah,Hirzebruch, in Essays on topology and related topics, 1970], where there isa reference to my result (but not to the paper itself). There is a similar referencein: [Bott and Taubes, Jour. Amer. Math. Soc. 2(1989)].

[10] (with J.Dupont) On manifolds satisfying w21 = 0, 1971

This paper was written in 1970 during my stay (1969-71) at IAS. It containsa generalization of the result on semicharacteristic in [8] where the orientabilityassumption w1 = 0 is weakened to w2

1 = 0 (w1 is the first Stiefel-Whitney class).This was used in the paper [Davis and Milgram, Trans. Amer. Math. Soc., 1989].The appendix of this paper is a study of the symmetric power SPnX where Xis a compact unorientable smooth 2-manifold whose first rational Betti number isg. We show that SPnX is a (2n − g)-dimensional bundle over a g-dimensionalrealtorus with fibre RP 2n−g, the real projective space of dimension 2n − g. Inparticular if X is a projective plane (g = 0) then SPnX = RP 2n; if X is a Kleinbottle (g = 1) then SPnX is a RP 2n−1-bundle over the circle. (SPnY for Y acompact Riemann surface was studied earlier in [Macdonald, Topology 1962].) Wealso show that S∞X is a product of a g-dimensional torus with RP∞. Our resultSPn(RP 2) = RP 2n is reinterpreted in [Arnold, Topological content of the Maxwelltheorem on multiple representations of spherical functions, Topological methodsin nonlinear analysis 7(1996)]. This paper contains also a study of a cobordismring G∗ based on closed manifolds together with an element Γ in H1(M,Z/4Z)which reduces to the first Stiefel Whitney class by reduction mod 2. Our explicitdetermination of this ring relies on earlier work of [C.T.C.Wall, Ann. Math. 1960];a special role in our description is played by the manifolds SPnX where X is aKlein bottle and n is a power of 2. (There are two natural choices for Γ but theyrepresent the same element in G∗.)

[11] Novikov’s higher signature andfamilies of elliptic operators, 1972

This paper was written in 1970 during my stay (1969-71) at IAS. I used it asmy Ph.D. thesis at Princeton University (may 1971). The main contribution ofthis paper is to introduce the analytic approach (based on the index theorem) toattack the Novikov’s conjecture on higher signature. That conjecture states that,if one multiplies the Hirzebruch L-class of a compact oriented manifold M with a

COMMENTS ON MY PAPERS 3

cohomology class which comes from the cohomology of the classifying space of thefundamental group of M and then one integrates the result over M , one obtainsa homotopy invariant of M . In this paper I introduce, for M of even dimensionwith fundamental group Zn, a family of elliptic operators on M . These operatorsare obtained by twisting the Atiyah-Singer signature operator by a variable flatvector bundle on M coming from a unitary one dimensional representation of thefundamental group. While the index of each of these operators is the same as thatof the untwisted operator it turns out the family of operators has an interestingindex in the K-theory of the parameter space of the space of flat bundles consideredand I showed that this index is on the one hand a homotopy invariant and on theother hand from it one can recover the whole Novikov higher signature thus provingNovikov’s conjecture in this case.

Another contribution of this paper is to formulate a version of the Hirzebruchsignature theorem in which cohomology is taken with coefficients in a flat vectorbundle with a flat hermitian form which is not necessarily positive definite. In thiscase Hirzebruch’s original proof (with constant coefficients) does not work but theAtiyah-Singer theorem can be used instead. In the paper I show that from this”twisted” signature theorem one can derive various examples where Novikov’s con-jecture holds for certain nonabelian fundamental groups. The analytic approachof this paper has been extended by Mischenko and Kasparov to the case wherethe fundamental group is a discrete subgroup of Lie groups and then by Connes,Moscovici, Gromov, Higson and others to even more general fundamental groups.See [Ferry,Ranicki,Rosenberg: Novikov signatures, index theory and rigidity, Lon-don Math.Soc.Lect.Notes, 1995] for a review of these developments. The twistedsignature theorem of this paper is used in: [Gromov and Lawson, Ann. Math.1980], [Atiyah, Math. Annalen 1987], [Gromov, in ”Functional analysis on the eveof the 21st century, II”, Progress in Math. 132,Birkhauser 1996].

[14]. Introduction to elliptic operators, 1974

This (mainly expository) paper is based on a lecture that I gave at a Tri-este summer school in 1972. It contains the definition of elliptic operators andtheir index. The only part which is perhaps non-standard is the definition ofanalytical index as a homomorphism ind : K(BT ∗M,ST ∗M) −→ Z where T ∗Mis the cotangent bundle of a compact manifold M , BT ∗M is its unit disk bun-dle, ST ∗M is its unit sphere bundle and K() is K-theory. The usual (Atiyah-Singer) definition of ind is via the theory of pseudo-differential operators. But inthis paper I show that if we are willing to increase Z to Z[1/2], one can defineind : K(BT ∗M,ST ∗M) −→ Z[1/2] in a more elementary way, using only differen-tial operators.

[17]. On the discrete series representations ofthe classical groups over a finite field, 1974

This paper represents my talk at the ICM in August 1974. I was originally

4 G. LUSZTIG

invited to give a talk in the Algebraic Topology section but I requested to changeto the Lie Groups section. In 1973 the only cases where cuspidal characters of areductive group over a finite field were constructed were GLn(Fq) (by J.A.Green),Sp4(Fq) (by B.Srinivasan) and G2(Fq) (by B.Chang, R.Ree). In 1973, after mystudy [13] of the ”Brauer lifting” of the standard n-dimensional representationof GLn(Fq), I tried to find the constituents of the Brauer lifting X of the stan-dard representation (of dimension N) of a symplectic orthogonal or unitary groupG(Fq). The result that I found is that X = X1 +X2 + ...+XN where Xi is ± anirreducible representation, X1 is up to sign a cuspidal representation (new at thetime) of dimension |G(Fq)| divided by the order of a ”Coxeter torus” and by theorder of a maximal unipotent subgroup; moreover Xi for i > 1 were noncuspidaland could be explicitly described as components of certain induced representationsfrom analogous cuspidal representations of Coxeter type of smaller classical groupsor GLn by determining explicitly the relevant Hecke algebras. Thus this methodgives a way to approach at least the ”Coxeter series” of cuspidal representationsof a classical group. This is what is explained in the first part of this paper.(The proofs of the results in the first part were never published since they weresuperseded by later developments.)

In the second part of the paper I described my joint work with Deligne (doneduring the spring 1974 at IHES) in which l-adic cohomology is used to con-struct representations of G(Fq) where G is a connected reductive group overFq. This method was first used by Tate and Thompson [Tate, Algebraic cy-cles and poles...,1965] who observed that the obvious action of the unitary groupU3(Fq) on the projective Fermat curve xq+1 + yq+1 + zq+1 = 0 over Fq inducesan action on H1 which is the (interesting) irreducible representation of degreeq2 − q of U3(Fq). Around 1973, Drinfeld observed that the cuspidal represen-tations of SL2(Fq) can be realized in the cohomology with compact support ofthe Dickson curve xyq − xqy = 1 over Fq by taking eigenspaces of the action ofT = t ∈ F ∗

q ; tq+1 = 1 which acts on the curve by homothety. (I learned about

this fact from T. A. Springer in 1973.) (Note that Dickson’s curve can be viewedas the (open) part of the Fermat curve where z 6= 0. This open part is stable underSL2(Fq)×T viewed as a subgroup of U3(Fq).) The main result of this section wasthe introduction for any element w in the Weyl group of G, of two new algebraicvarieties: the variety Xw of Borel subgroups B of G such that B and its imageunder the Frobenius map F are in relative position w and the finite principal cov-ering Xw of Xw whose group is the group Tw of rational points of an F -stabletorus of type w of G. (Note that the Drinfeld curve is a special case of Xw in thecase G = SL2 and the Tate-Thompson curve is the compactification of an Xw inthe case G = GL3 with a nonsplit Fq-structure.) These varieties admit naturalaction of G(Fq) and the principal covering above gives rise to G(Fq)-equivariantlocal systems on Xw, one for each character θ of Tw. By passing to cohomol-ogy with compact support with coefficients in such a local system one obtainsrepresentations of G(Fq). By taking alternating sums one obtains certain virtual


representations R(w, θ) of G(Fq) indexed by the various θ. At the time whenthis paper was written (summer 1974) we conjectured that R(w, θ) for θ genericare up to sign irreducible characters which provide a solution of the Macdonaldconjecture.

[18] Sur la conjecture de Macdonald, 1975

By my joint work with Deligne (spring 1974) described in [17], a conjecturalsolution to the Macdonald conjecture for the irreducible representations of G(Fq)was known in terms of the the virtual representations R(w, θ) defined by the sub-

varieties Xw of the flag manifold, their finite coverings Xw and their cohomologywith compact support. But it was not clear how to prove the irreducibility ofR(w, θ) for θ generic or how to compute the degree of R(w, θ). In the fall 1974 (atWarwick) I completed the proof of the fact that the virtual representations R(w, θ)defined in [17] are indeed a solution to the Macdonald conjecture. In this paper(written in late 1974) I sketched this proof; it is based on the following principle:

(∗) Assume that H is a finite group acting on an algebraic variety Y in such away that the space of orbits Y/H is again an algebraic variety Y ′; assume furtherthat there is a partition of Y into finitely many locally closed H-stable pieces Yi

and on each Yi the action of H extends to an action of a connected algebraic groupHi. Then Y ′, Y have the same Euler characteristic.

The main observation of this paper is that (∗) is applicable in the following twocases:

(A) Y = Xw/P (Fq), Y′ = Xw/P (Fq) where P is a parabolic subgroup defined

over Fq;

(B) Y = (Xw ×Xw′)/G(Fq), Y′ = (Xw × Xw′)/G(Fq), where w,w′ are Weyl

group elements.Now (∗) in case (A) implies easily the expected formula for the degree of R(w, θ)and (∗) in case (B) implies easily an explicit formula for the inner product of anR(w, θ) with an R(w′, θ′) from which the desired irreducibility result follows. Theproof of the degree formula and that of the inner product formula given later in[22] are quite different: they use a disjointness theorem.

[19] Divisibility of projective modules of finiteChevalley groups by the Steinberg module, 1976

This paper was written during my stay at IHES in the spring of 1974. Themotivation for this paper was to find evidence for the Macdonald conjecture for aconnected reductive group G defined over Fq. (However, by the time this paperappeared, Macdonald’s conjecture was proved.) If T is a maximal torus of Gdefined over Fq and θ is a character of T (Fq) then the induced representation R =

IndG(Fq)

T (Fq)(θ) is defined. If θ is in general position and if R(T, θ) is the irreducible

representation of G(Fq) provided by Macdonald’s conjecture (assumed to hold)then R is isomorphic to R(T, θ) ⊗ S where S is the Steinberg representation.

6 G. LUSZTIG

Therefore if we can prove apriori that R is isomorphic to S tensor some virtualrepresentation, then this would be evidence for the Macdonald conjecture. NowR can be viewed as a representation coming from a projective G(Fq)-module overthe ring of integers in a suitable p-adic field. Hence it would be enough to showthat any such projective G(Fq)-module is ”divisible” by S. This is what is shownin this paper.

[20] A note on nilpotent matrices of fixed rank, 1976

This paper was written in early fall 1974. At that time the series of represen-tations of a reductive group attached to a maximal torus over Fq were alreadyconstructed (in the joint work with Deligne, see [17]) but their irreducibility wasnot yet proved (it was proved shortly afterwards in [18]). But in the case of theeven nonsplit orthogonal group over Fq and the series corresponding to the Cox-eter torus the character was explicitly computable (in this case I could computethe Green functions, by some computations which later became part of [23]) hencein this case irreducibility could be proved directly by using orthogonality of theexplicit Green functions; to do this it was necessary to know the number of unipo-tent elements u such that u− 1 has fixed rank. (It turned out out that the Greenfunction on a unipotent element u depended only on the rank of u − 1.) Thisnumber is computed in this paper. The result of this paper gives a new proof (forclassical groups over Fq) for Steinberg’s theorem on the total number of unipotentelements. (For GLn that theorem is due to Fine and Herstein and independentlyto Ph.Hall.)

[22] (with P.Deligne) Representations ofreductive groups over finite fields, 1976

This paper (written during the first half of 1975) contains a detailed study

of the varieties Xw, Xw associated in [17] (by me and Deligne) to an element win the Weyl group of a connected reductive group G defined over a finite fieldFq and of the associated virtual representations R(w, θ) of G(Fq). Here θ is acharacter of the finite ”torus” Tw of type w. In Section 3 a Lefschetz type fixedpoint formula for a transformation of finite order of an algebraic variety is given.(This formula was already used implicitly in [17].) This is used in Section 4 toprove a formula for the character of R(w, θ) assuming that the Green functionsof G and smaller groups are known. In Section 6 the disjointness theorem isproved: the virtual representations R(w, θ), R(w′, θ′) are disjoint unless θ, θ′ areconjugate after extension of the ground field (and composition with the trace);it is also shown that the equivalence classes of various θ as above can be viewedas semisimple conjugacy classes defined over Fq in the ”dual” group G∗ (at leastif G has connected centre). The proof of the disjointness uses the possibility of

extending the action of a finite group on (Xw × Xw′)/G(Fq) to the action of ahigher dimensional group, not on the whole variety, but separately on each piece


of a partition of the variety in pieces stable under the finite group (compare with(∗) in the comments to [18]). The disjointness theorem has several applications(which were proved in a different way in [18]): the degree formula for R(w, θ); theinner product formula for R(w, θ), R(w′, θ′); the orthogonality formula for Greenfunctions. In Corollary 7.7 it is shown that any irreducible representation of G(Fq)appears in some R(w, θ); a completely different proof of this result based on thetheory of perverse sheaves was later given in [178]. Combining this result withthe disjointness theorem one obtains a canonical map from the set of irreduciblerepresentations of G(Fq) (up to isomorphism) to the set of semisimple conjugacyclasses in G∗(Fq) (at least if G has connected centre); this is an initial but crucialstep in the classification of irreducible representations of G(Fq). In Section 9 itis shown that Xw is affine assuming that q is greater than the Coxeter number.Recently it has been shown that Xw is affine without restriction on q assumingthat w has minimal length in its twisted conjugacy class [Orlik and Rapoport,J. Algebra, 2008] and [He, J. Algebra, 2008], see also [Bonnafe and Rouquier, J.Algebra, 2008]. In 9.16 it is shown that any Green function evaluated at a regularunipotent element is equal to 1. In Section 10 (assuming that the centre of G isconnected) it is shown how to parametrize explicitly the irreducible components ofthe Gelfand-Graev representations and that these components are explicit linearcombinations of R(w, θ). In Section 11 the results of the paper are extended toRee and Suzuki groups. In this case the inner product formula cannot be handledby the methods of this paper in the case q =

√2 or q =

√3, but it can be handled

by the proof given later in [30].

[23] On the Green polynomials of classical grouyps, 1976

This paper was written in the summer of 1975, after the completion of [22]. In[22] a general study of the variety Xw (all Borel subgroups in relative position wwith their transform under Frobenius in a reductive group G over Fq) was made.In the present paper I tried to study in detail the first nontrivial class of examplesof the variety Xw, namely the case where G is a classical group and w is a Coxeterelement of minimal length. In this case I obtained explicit formulas for (a) thenumber of rational points of Xw over any extension of the ground field and for (b)the Green functions (alternating sums of traces of a unipotent element of G(Fq)on the cohomology with compact support of Xw. In the case of symplectic groups,(b) solved a conjecture of B.Srinivasan. This paper was a preparation for my nextproject [25] in which I studied Xw for w a (twisted) Coxeter element of minimallength for a general G.

[24] On the finiteness of the number of unipotent classes, 1976

The work on this paper was started during a visit to IHES in December 1974and was completed during another visit to IHES in December 1975. Let G bea connected reductive group defined over Fq, let P be a parabolic subgroup ofG (not necesarily defined over Fq) and let L be a Levi subgroup of P defined

8 G. LUSZTIG

over Fq. In this paper I define for any virtual representation ρ of L(Fq) a virtualrepresentation R(L, P, ρ) of G(Fq). In the case where P is defined over Fq thisis the usual induced representation from P (Fq) to G(Fq) where ρ is viewed as avirtual representation of P (Fq). In the case where L is a maximal torus definedover Fq and ρ is a character of T (Fq), this reduces to the construction in [22].One of the main results of this paper is an inner product formula for two virtualrepresentations R(L, P, ρ), R(L′, P ′, ρ′) under a genericity assumption. One con-sequence of this is that, under a genericity assumption, R(L, P, ρ) is irreducible(up to sign) for ρ irreducible. This irreducibilty result plays a key role in my laterpapers [29],[57]; it allows one to construct new irreducible representations startingwith known irreducible representations of L(Fq). Another consequence is that thenumber of unipotent conjugacy classes in G is finite, answering a conjecture ofSteinberg at the ICM in 1966. Previously this finiteness result was known only forclassical groups or for exceptional groups in good characteristic. A few years later(1980) Mizuno gave another proof of the finiteness result for exceptional groups inbad characteristic based on extensive computations. I believe that the proof givenin this paper is still the only proof of finiteness which does not use classification.After this paper was written, Deligne stated a refinement of the inner productformula for R(L, P, ρ), R(L′, P ′, ρ′) (without genericity assumptions) as an ana-logue of Mackey’s theorem (generalizing the case where L, L′ are maximal tori,known from [22]) and proved it assuming that q is large. Deligne’s proof remainsunpublished. Around 1985 I proved a character sheaf version of this formula (see[65]); in view of the results of [89] this implies the refined inner product formulafor representations (again for large q). A version of Deligne’s proof appeared in[Bonnafe, J.Alg. 1998]. Recently [Bonnafe and Michel, J. Alg., 2011] gave a proofof this formula with a very mild assumption on q, using computer calculation. Thegeneral case is still not proved. The (refined) inner product formula would implythat R(L, P, ρ) is independent of P .

[25] Coxeter orbits and eigenspaces of Frobenius, 1976

The work on this paper was done in late 1975. This paper continues the projectstarted in [23] to study in detail the variety Xw of [22] in the case where w is a(twisted) Coxeter of minimal length in the Weyl group of a connected reductivealmost simple group G defined over Fq. (The case of Suzuki and Ree groups isalso treated in the paper.) One of the main results of this paper is a constructionof several new unipotent cuspidal representations of G(Fq) in the case where G isexceptional. Let d be the smallest integer ≥ 1 such that Xw is stable under F d,the d-th power of the Frobenius map. In this paper I give

(a) an explicit computation of the eigenvalues of F d on H∗c (Xw) and an explicit

formula for the dimensions of its eigenspaces;(b) a proof that F d acts semisimply on H∗

c (Xw) and its eigenspaces are irre-ducible, mutually nonisomorphic G(Fq)-modules.In the case where G is of type E7 we have d = 1 and two of the eigenvalues


of F are of the form√

−q7, providing the first examples in a split case whereF : H∗

c (Xw) −→ H∗c (Xw) can have eigenvalues with absolute value not an integer

power of q. In the case where G is a Suzuki or Ree group of type B2 or G2 withq =

√Q (Q is an odd power of m, m = 2 or 3), the variety Xw for w a simple

reflection is an affine curve and its compactification Xw is a smooth projectivecurve defined over FQ such that Xw −Xw consists of Qm+1 points. On the otherhand by theorem 3.3(i) of this paper, Xw has no FQ-rational points. It followsthat the number of FQ-rational points of Xw ie equal to Qm + 1. Note also thatthe genus of Xw is determined explicitly from (a) or [22]. In a letter to me datedMay 11, 1983, J.-P.Serre made the following remarks.

(1) Xw has the maximum number of FQ-rational points compatible with itsgenus.

(2) If a smooth curve over FQ has Qm + 1 rational points and has the samegenus as Xw then it has the same zeta function as Xw (which is determined from(a)).Due to property (1), these curves have been used to produce Goppa (error cor-recting) codes. See [N. Hurt: Many rational points; coding theory and algebraicgeometry, Kluwer, 2003].

[29] Irreducible representations of finite classical groups, 1977

The work on this paper was started at Warwick during the summer of 1976 andcompleted during my visit to MIT in the fall 1976. This paper contains the classi-fication and degrees of the irreducible complex representations of classical groups(with connected centre) other than GLn, over a finite field. It relies on: the useof the ”cohomological induction” [22],[23]; the use of the dimension formulas forthe irreducible representations of Hecke algebras of type B with two parameters[Hoefsmit, UBC Ph.D. Thesis, 1974] (of which I learned from B.Chang during myvisit to Vancouver at ICM-1974). This paper establishes what was later called ”theJordan decomposition” for the representations of classical groups (with connectedcentre). It also establishes the parametrization of unipotent representation forthese groups in terms of some new combinatorial objects, the ”symbols” and theclassification of unipotent cuspidal representations of classical groups (for exampleSp2n(Fq) has such a representation if and only if n = k2+k which is then unique).Also it is shown that the endomorphism algebra of the representation inducedfrom a unipotent cuspidal (or more generally isolated cuspidal) representation toa larger classical group is an Iwahori-Hecke algebra (anticipating a later result of[Howlett and Lehrer, Inv. Math. 1980]) and giving also precise information on thevalues of the parameters of that Iwahori-Hecke algebra. For this we need to countin terms of generating functions the number of conjugacy classes in a classicalgroup with connected centre. This together with an inductive hypothesis and themethods outlined above give a way to predict the number of isolated cuspidal rep-resentations. The degrees of these isolated cuspidal representations can be guessedusing the technique of symbols by ”interpolation” from the degrees of noncuspi-

10 G. LUSZTIG

dal representations. To prove that these guessed are correct we need to calculatethe sum of squares of the (guessed) degrees of unipotent representations which isperhaps the most interesting part of this paper. To do this I find explicit formulas(for each irreducible representation E of the Weyl group) of the polynomial dE(q)whose coefficients record the multiplicities of E in the various cohomology spacesof the flag manifold. (I do this first for GLn and then reduce the case of classicalgroups to that of GLn.) Then I show that the (guessed) degree polynomials can beexpressed as linear combinations of the dE(q) with constant coefficients of the formplus or minus 1/2s. This anticipates the notion of family of representations of theWeyl group and the role of the nonabelian Fourier transform [34] (which in thiscase happens to be abelian.) Here the use of the technique of symbols (introducedin this paper) is crucial. It is remarkable that suitable variations of the notion ofsymbol (used here in connection with unipotent representations) were later shownto be exactly what one needs to describe explicitly the Springer correspondence(including the generalized one) for classical groups [59],[61] and for classical Liealgebras in characteristic 2 [T. Xue, 2009].

[30] Representations of finite Chevalley groups, 1978

This paper represents lectures that I gave in August 1977 at Madison, Wiscon-sin. Let G be a connected reductive group defined over Fq. Among other things, inthis paper I give two refinements (see (a),(b) below) of the inner product formulafor the virtual representations R(w, θ) (see [22]) of G(Fq):

(a) a proof (in 2.3) which applies equally well to the Ree and Suzuki groups

with q =√2 or q =

√3 which were not covered by earlier proofs in [18],[22];

(b) a proof (see 3.8) of the fact that, if w,w′ are in the Weyl group W andXw, Xw′ are the varieties of [22], then |((Xw ×Xw′)/G(Fq))(Fqs)| is equal to thetrace of a linear transformation h 7→ TwhTw′−1 of the Hecke algebra with parameterqs. (Assume that G is split over Fq.)Note that (b) is a refinement of the inner product formula for R(w, θ) (in thecase where θ = 1) since for that formula one needs the Euler characteristic of(Xw ×Xw′)/G(Fq) which is the limit of |((Xw ×Xw′)/G(Fq))(Fqs)| as s goes to0. I also show that to any unipotent representation of G(Fq) one can attach aneigenvalue of Frobenius well defined up to an integer power of q. This result waslater used in [Digne and Michel, C.R. Acad. Sci. Paris, 1980] and by [Asai, OsakaJ.Math., 1983]. In 3.34 these eigenvalue of Frobenius are described in all casesarising in type 6= E8. On page 26 (see (d)) it is shown that Xw is irreducible ifand only if for any simple reflection s, some element in the Frobenius orbit of sappears in a reduced expression of w; this result has been rediscovered in [Bonnafeand Rouquier, C.R. Acad. Sci. Paris, 2006]. (Another proof is given in [Goertz,Repres.Th., 2009]). In 3.13 and 3.16 an explicit formula for the sum of squares ofunipotent representations of G(Fq) is given. This is used in 3.24 to classify theunipotent representations in the case where G is split of type E6 or E7 (it turnsout that all cuspidal unipotent representations arise from the analysis in [25]). In


the case where G is nonsplit E6, triality D4 or F4, a classification of unipotentrepresentations is again given assuming that q is large; in these cases there arecuspidal unipotent representations which do not arise from the analysis in [25].On page 24 (see (b)) (assuming that Frobenius acts on W as conjugation by thelongest element w0) I define for each w ∈ W a bijective morphism tw : Xw0

−→Xw0

as follows: tw(B) = B′ where B ∈ Xw0and B′ is defined by pos(B,B′) =

w, pos(B′, F rob(B)) = w−1w0 (so that B′ ∈ Xw0); I show that the tw define a

homomorphism of the braid group in the group of permutations of Xw0. Moreover

I show that after passage to cohomology one obtains a representation of the Heckealgebra of W with parameter −q on H∗

c (Xw0). Several years later (around 1982)

I used a similar idea in the case where G is of type D4 so that W has simplereflections s0, s1, s2, s3 with s1, s2, s3 commuting and w = s1s0s2s0s3s0 ∈ W .(This is unpublished but there is a reference to it in [Broue and Malle, Asterisque1993, 5A] and [Broue and Michel, Progr.in Math.141, 1997, page 114].) Let Z(w)be the centralizer of w in W . Namely, I defined three permutations A,B,C ofXw into itself (similar to tw above) such that A,B,C commute with Frobeniusand ABC = BCA = CAB = Frobenius. The maps A,B,C are associated tothree generators a, b, c of Z(w) which satisfy abc = bca = cab = w. This suggeststhat the ”braid group” corresponding to Z(w) (a complex reflection group) shouldhave the relation ABC = BCA = CAB. Indeed, this later appeared as a specialcase of the relations of such ”braid groups” given in [Broue and Malle, Asterisque1993]. The idea in this example was further pursued in [Digne and Michel, NagoyaMath.J.,2006] and in [203].

[31] (with W.M.Beynon) Some numerical results onthe characters of exceptional Weyl groups, 1978

For any irreducible representation E of a Weyl group, the fake degree dE(q) ofE is defined in [30] as the polynomial in q which records the multiplicities of E inthe various cohomology spaces of the flag manifold. After the polynomials dE(q)were explicitly computed in [29] in the case of classical groups, it was naturalto try to compute them for simple exceptional groups. This is what is done inthe present paper, using a computer and the known character tables of W (butwe found and corrected some errors in the character table for type E8). Thesecomputations were later used in [34]. One observation of this paper is that thepolynomials dE(q) are palindromic apart from a small number of exceptions intype E7 (for E of degree 512) and E8 (for E of degree 4096). My collaborator,W.M.Beynon, was a computer expert at Warwick; I was introduced to him nyR.W.Carter.

[33] On the reflection representationof a finite Chevalley group, 1979

The work on this paper was done in the spring of 1977; the results were presentedat an LMS Symposium on Representations of Lie Groups in Oxford (July 1977).

12 G. LUSZTIG

I will explain the main result of this paper using concepts which were developedseveral years after the paper was written (theory of character sheaves). Let G bea connected reductive group over an algebraic closure of a finite field Fq with afixed Fq-split rational structure and Frobenius map F : G → G. For each w in theWeyl group one can consider (following [22]) the variety Xw of Borel subgroups Bof G such that B, FB are in position w. Then G(Fq) acts naturally on the l-adiccohomology Hi

c(Xw). Replacing F by conjugation by an element g ∈ G one canconsider the variety Yw,g of Borel subgroups B of G such that B, gBg−1 are inposition w. The union over g in G of these varieties maps naturaly to G and we cantake the direct image Kw with compact support of the sheaf Ql under this map.Then pHiKw are perverse sheaves on G. Now for any irreducible representation Eof the Weyl group we denote by Eq the corresponding irreducible representation ofG(Fq) which appears in H0

c (X1) (functions on the flag manifold of G(Fq)) and wedenote by E1 the simple perverse sheaf on G corresponding to E which appearsin K1 (a perverse sheaf on G up to shift, with W -action). The main result of thispaper is that for any w we have

∑

i

(−1)i(Eq : Hic(Xw) = (−1)dimG

∑

i

(−1)i(E1 : pHiKw)

where (:) denotes multiplicity. Here the left hand side can be interpreted as thevalue of the character of Eq on a regular semisimple element in a maximal torus oftype w. This result does not compute these character values but it shows that thesevalues are universal invariants which make sense also over complex numbers. Nowthe multiplicity (E1 : pHiKw) does not change when E1,

pHiKw are restrictedto the variety of regular semisimple elements of G. But after this restrictionE1,

pHiKw become local systems and (E1 : pHiKw) is equal to the correspondingmultiplicity of local systems which can be considered independently of the theoryof perverse sheaves. It is in this form that the result above appears in the presentpaper where the varieties Yw,g are introduced only for g regular semisimple (inwhich case they are shown to be smooth of dimension equal to the length of w).Thus this paper can be viewed as a precursor of the theory of character sheaveswhich was developped in [63-65,68,69]. As an application I determine explicitlythe value of the character of the ”reflection representation” of G(Fq) (constructedearlier by Kilmoyer) on a regular semisimple elements of type w assuming that Gis of type A,D, or E. Namely, it is shown that this value is equal to the trace ofw on the reflection representation of W . At the time when the paper was writtenthis result was new for types D,E.

[34] Unipotent representations of afinite Chevalley group of type E8, 1979

This paper was written in the spring of 1978, soon after my arrival to MIT(January 1978). This paper introduces a new type of Fourier transform (the ”non-abelian Fourier transform”). It is a unitary involution of the vector space of


functions on a set M(G) associated to a finite group G; here M(G) is the set of allpairs (x, r) where x is an element of G (up to conjugacy) and r is an irreduciblerepresentation of the centralizer of x (up to isomorphism). About ten years later:

-I found [77] an interpretation of the ”non-abelian Fourier transform” as the”character table” of the equivariant complexified K-theory convolution algebraKG(G) (where G acts on itself by conjugation): this (commutative) algebra has anatural basis indexed by M(G) and its (one dimensional) representations are alsoindexed by M(G) hence its character table is defined;

-physicists [Dijkgraaf, Vafa, E.Verlinde, H.Verlinde, Comm. Math. Phys. 1989],[Dijkgraaf, Pasquier, Roche, Nuclear Phys. 1990] rediscovered this Fourier trans-form (possibly with a twist by a 3-cocycle);

-Drinfeld explained it in terms of his ”double” of the group algebra of G (hetold me about this around 1990).In this paper, the ”non-abelian Fourier transform” is used to complete the clas-sification and computation of degrees of the unipotent representations of finiteChevalley groups (started in [29,30]). Note that for the analogous problem forclassical groups, the standard (abelian) Fourier transform is sufficient.

It is remarkable that the ”non-abelian Fourier transform” enters in an essentialway in subsequent works in representation theory: the multiplicity formulas in thevirtual representations R(T, θ) of [22], see [57]; the analogous multiplicity formulasfor character sheaves [63-65,68,69]; the relation of character sheaves to irreduciblecharacters, see [71,102] and [Shoji, Adv.Math.1995]. This paper (see Section 8)also introduces the concept of ”special representation” of a Weyl group (which wasfurther developed in [36]) and that of ”family” of unipotent representations (whichcontains as a particular case the notion of family of irreducible representations of aWeyl group). The concept of special representation of a Weyl group was suggestedby the calculations in [29]. It is nowadays used extensively in the representationtheory of reductive groups over real or complex numbers.

[35] (with N.Spaltenstein) Induced unipotent classes, 1979

Let G be a connected reductive group over an algebraically closed field, let Lbe a Levi subgroup of a parabolic subgroup P of G and let C be a unipotentclass of G. In this paper we associate to L,C a unipotent class C′ of G (said tobe induced by C); it is the unique unipotent class of G whose intersection withCUP is dense in CUP (here UP is the unipotent radical of P ). In this paperwe show that C′ does not depend on the choice of P and that C′ ∩ CUP is asingle P -conjugacy class. When C = 1, then C′ is the Richardson class definedby L. We give two proofs for the independence on P ; one of these depends onsome results on representations of a reductive group over a finite field and on theLang-Weil estimates; the other is more elementary but uses some case by casearguments. In this paper we also introduce the idea of truncated induction forrepresentations of Weyl groups generalizing a construction of Macdonald. Weshow that the Springer representation attached to an induced unipotent class

14 G. LUSZTIG

is obtained from the Springer representation of the original unipotent class bytruncated induction. This has been used in subsequent works (such as [36, 48])to compute the Springer correspondence in certain cases arising from exceptionalgroups.

[36] A class of irreducible representations of a Weyl group, 1979

In this paper (written in the summer of 1978) I give an alternative definitionof the class SW of irreducible representations of a Weyl group W of a complexadjoint group G (introduced in [34] and later called ”special representations”).We have two commuting involutions A,B of Irr(W ): A is tensoring by the signrepresentations and B is the q = 1 specialization of an involution of the set ofirreducible representations of the Hecke algebra given by the action of the Galoisgroup which takes

√q to −√

q. (Note that B is the identity for W of classical typeand it is almost the identity in general.) Let T = AB = BA. In this paper it isshown that

(i) SW is preserved by the truncated induction [35] from a parabolic subgroup,and

(ii) SW is preserved by T ;moreover, SW is characterized by (i)-(ii) and the fact that it contains the unitrepresentation. This result has the following consequence for the two-sided cells(introduced later in [37]) of W . Let w0 be the longest element of W and let c bea two-sided cell of W (we assume W 6= 1). Then c′ := cw0 = w0c is again a twosided cell and either c or c′ meets a proper parabolic subgroup of W . (This kindof result allowed me (in the later work [57]) to analyze unipotent representationsinductively by using ”truncated induction” from a proper parabolic subgroup and”duality” (which interchanges c, c′).) The class SW is explicitly computed in eachcase (using the formalism of symbols [29] for classical types), the results of [31] on”fake degrees” and the results of [35] on Springer representations. In this paper(Sec.9) I formulate the idea of ”special unipotent class” of G (although I did notuse the word ”special”): these are unipotent classes in 1-1 correspondence withthe special representation of W (under the Springer correspondence). Since the setof special representations of W admits a natural involution (given by T above) itfollows that the set of special unipotent classes admits a natural involution. (Forexample the class 1 is interchanged with the regular unipotent class.) Later,[Spaltenstein, LNM 946, III, Springer Verlag 1982], motivated by this paper (asmentioned in [loc.cit., p.210]) gave a definition of a subset of the set of unipotentclasses of G and an order reversing involution of this subset; this definition is basedon properties of the partial order of the set of unipotent classes and is somewhatunsatisfactory [loc.cit., p.210] for exceptional types. One can show that the subsetdefined in [loc.cit.] is the same as the set of special unipotent classes but this wasstated in [loc.cit.] not as a fact but as an analogy. In this paper I also define aclass SW of irreducible representations of W which contains SW , but unlike SW ,depends on the underlying root system. The representations in SW are obtained


by truncated induction [35] from special representations of subgroups of W whichare Weyl groups of Borel-de Siebenthal subgroups (=centralizers of semisimpleelements) of the dual group G∗ of G. I believe that the most interesting andunexpected contribution of this paper is the statement that SW is in bijectionwith the set of unipotent classes in G via Springer’s correspondence when G is ofclassical type and conjecturally in general; for exceptional groups this was verifiedin [48], see also 13.3 of [57]. (The details of the proof for classical groups appearedonly 25 years later in [188].) This has the following consequence: there is a naturalmap from the set of special unipotent classes of a Borel-de Siebenthal subgroupof G∗ (or its dual) to the set of unipotent classes in G; moreover, all unipotentclasses in G appear in this way. This map has been later interpreted in termsof representation theory in [Barbasch and Vogan, Primitive ideals and orbitalintegrals..., Math. Ann. 1982] (for complex groups) and in [100] (for groups overFq and character sheaves).

[37] (with D.Kazhdan) Representations ofCoxeter groups and Hecke algebras, 1979

The work on this paper was done in late 1978 and early 1979. My motivationfor this work came from the desire to construct explicitly representations of theHecke algebra H with parameter q and standard basis Tw;w ∈ W attached toa Weyl group W . A basis B of a vector space with W -action is said to be ”good”if for any simple reflection s of W and any b ∈ B we have either sb = −b orsb = b+

∑

b′∈B;b′ 6=b,sb′=−b′ ab,b′,sb′ where ab,b′,s are integers. Similarly, a basis B of

a vector space withH-action is said to be ”good” if for any simple reflection s ofWand any b ∈ B we have either Tsb = −b or Tsb = qb+

√q∑

b′∈B;b′ 6=b,Tsb′=−b′ ab,b′,sb′

where ab,b′,s are integers. In late 1977 I showed that if u is a unipotent element in asemisimple group G over C and Bu is the variety of Borel subgroups containing u,then in the Springer representation of W on Htop(Bu) the basis given by the irre-ducible components of Bu is good with ab,b′,s ≥ 0. This appeared in a letter I sentto Springer (March 1978). The same idea appeared in [Hotta, J. Fac. Sci. Univ.Tokyo, 1982] where the letter above is cited. In the case where u is subregular(with G of type ADE), Htop(Bu) could be identified with the reflection repre-sentation of W with the basis formed by simple roots. In Kilmoyer’s MIT thesis(which became a part of [Curtis, Iwahori, Kilmoyer, Publ. Math. IHES, 1971])an explicit q-deformation of the reflection representation of W to a representationof H with a good basis is found. This suggested that the bases of Htop(Bu) (asabove) may admit a q-analog which are good bases for an H-action. This is so forfor G = SL4, SL5. One of the main results of this paper is the definition of a newbasis cw;w ∈ W of H (with q an indeterminate) which is good for the left andright action of H on H. I will try to explain the definition of the elements cw in away somewhat different from the paper. Let w0 be the longest element of W . Wehave TsTw0

= qTsw0+ (q − 1)Tw0

. Here the right hand side has some coefficientq − 1 but if you replace Ts by Ts + 1 we obtain (Ts + 1)Tw0

= qTsw0+ qTw0

and

16 G. LUSZTIG

now the right hand side has only coefficients q.Assume that W is of type A3 with generators 1, 2, 3. We have

T2132Tw0= (q4 − 3q3 + 4q2 − 3q + 1)T312312

+ (q4 − 3q3 + 3q2 − q)T13213 + (q4 − 3q3 + 3q2 − q)T32312

+ (q4 − 3q3 + 3q2 − q)T12312 + (q4 − 2q3 + q2)T3212 + (q4 − 2q3 + q2)T1232

+ (q4 − 2q3 + q2)T2312 + (q4 − 2q3 + q2)T3213 + (q4 − 2q3 + q2)T1213

+ (q4 − q3)T213 + (q4 − q3)T123 + (q4 − q3)T321 + (q4 − q3)T312 + q4T13

and again the right hand side has several coefficients involving powers of q far fromq4. As in the case of Ts we can hope that by adding to T2132 a linear combinationof the Ty (with y strictly less than 2132) with coefficients sums of powers of qvery close to 1, the resulting sum times Tw0

is a linear combination of Ty′ withcoefficients sums of powers of q very close to q4. There is a unique way to that:

(T2312 + T231 + T232 + T312 + T212 + T23 + T32 + T12 + T21

+ T13 + T1 + T3 + (q + 1)T2 + (q + 1))Tw0

= (q4 + q3)T312312 + (q4 + q3)T13213 + q4T32312 + q4T12312

+ q4T3212 + q4T1232 + q4T2312 + q4T3213 + q4T1213 + q4T213

+ q4T123 + q4T321 + q4T312 + q4T13.

We then take

c∗2312 = T2312 + T231 + T232 + T312 + T212 + T23

+ T32 + T12 + T21 + T13 + T1 + T3 + (q + 1)T2 + (q + 1).

This procedure works in general and leads to a basis c∗w;w ∈ W of H. Moreexplicitly, c∗w =

∑

y≤w Py,w(q)Ty is characterized by

c∗wTw0=

∑

y≤w

ql(w)P ′y,w(q

−1)Tw0y

where Py,w, P′y,w are polynomials in q of degree at most (l(w)− l(y)−1)/2 if y 6= w

and Pw,w = P ′w,w = 1.) Let (cw) be the basis obtained from c∗w by the involution

Ts → −qT−1s of H.

In this paper it is shown that the basis (cw) is good for both the left and rightH-module structure on H (and in fact the scalar ab,b′,s is independent of s wheneverit is nonzero. Note also that the definition of c∗w is applicable to any Coxeter groupby replacing the operation of multiplication by Tw0

by the bar operation whichreplaces Tx by T−1

x−1 and q by q−1 (which is what appears in the paper).


Also in the paper left cells, right cells and two sided cells are introduced for anyCoxeter group and the left cells in type A are determined explicitly. The inversionformula 3.1 shows that the inverse of the triangular matrix (Py,w) is the triangularmatrix which has again the entries Py,w in another indexing and with some signchanges. This inversion formula was later generalized in [Vogan, Duke Math.J.1982] to the case of symmetric spaces, in which case a passage to the Langlandsdual of G is necessary. In this paper it is observed that the nontriviality of Py,w isvery closely related to the failure of local Poincare duality on a Schubert variety.The fact that the equivalence relation on the set of irreducible representations ofW given by the two-sided cells (of this paper) seemed to coincide with the equiva-lence relation defined by the families (introduced earlier in [34] in conection withthe representation theory of finite reductive groups), suggested that the Py,w mayhave a representation theory significance. In this paper, a conjecture is stated tothe effect that Py,w(1) should be equal to the multiplicity [Ly : Mw] of a sim-ple highest weight module Ly in a Verma module Mw over the Lie algebra of G.Some evidence for the conjecture (in addition to the one mentioned above) camefrom the fact that the matrix [Ly : Mw] was known in the literature for rank ≤ 3(Jantzen) and the Py,w could be explicitly computed in rank ≤ 3 and they matchedthe [Ly : Mw]. Another evidence came from [Joseph, W-module structure on theprimitive spectrum...,1979] which showed among other things that the basis of theregular representation of W given by

∑

y∈W sign(y)sign(w)[Ly : Mw]y;w ∈ Wis good. A bridge between the two sides of the conjecture was established in [39]via local intersection cohomology. As a consequence, the conjecture became theequality between [Ly : Mw] and the Euler characteristic of a certain local intersec-tion cohomology space. In this form the conjecture was established in [Beilinsonand Bernstein, C.R. Acad. Sci. Paris, 1981] and [Brylinski and Kashiwara, Invent.Math. 1981]. Later I formulated an extension of the conjecture above to expressthe character of ireducible highest weight modules with positive central charge ofany affine Lie algebra which involved the values at 1 of the entries of the matrixinverse to (Py,w); I communicated this conjecture (and also the similar conjecturefor any Kac-Moody Lie algebra) to V.Kac and the conjecture appeared in [De-odhar, Gabber and Kac, 1982]. (A proof was given in [Kashiwara and Tanisaki,Grothendieck Festschrift II, 1990].) Even later [88], I formulated an extension ofthe conjecture above to express the character of ireducible highest weight moduleswith negative central charge of any affine Lie algebra which involved the values at1 of the Py,w themselves. (A proof was given in [Kashiwara and Tanisaki, DukeMath.J., 1995].)

A generalization of the notion of left/right/two-sided cells of this paper to thecase of complex reflection groups has been proposed in [Bonnafe and Rouquier,Cellules de Calogero-Moser, arxiv:1302.2720]; earlier, [Gordon and Martino, Math.Res. Lett. 16(2009)] proposed a generalization to complex reflection groups of thenotion of family of irreducible representations of a Weyl group introduced in [34].

18 G. LUSZTIG

[38] (with D.Kazhdan) A topologicalapproach to Springer’s representations, 1980

This paper was written in 1979 but the work on it was done in late summerof 1978 (except Sec.7). In 1976, Springer defined an action of the Weyl groupW on the cohomology H∗(Bu) of the variety Bu of Borel subgroups containinga unipotent element u of a reductive algebraic group over C, using methods incharacteristic p > 0. Moreover in a letter to me (1977) Springer defined an actionof W×W on the cohomology H∗

c (Z) of the Steinberg variety Z of triples (u,B,B′)where u is a variable unipotent element and B,B′ are Borel subgroups containingu; in the same letter he conjectured that the representation in Htop

c (Z) is the bi-regular representation. In this paper an elementary construction of the Springerrepresentation of W on Htop(Bu) and of the Springer representation of W ×W onHtop

c (Z) is given and the conjecture of Springer mentioned above is proved. Theconstruction in this paper is based on an explicit homotopy equivalence si from Bu

to Bu for any simple reflection si in W . We were expecting (but unable to prove)that the maps si give a representation of W in the group of homotopy equivalencesmodulo homotopy of Bu; we could only prove this after passage to cohomologyand only in top degree. The stronger statement has been established later in[Rossmann, J. Funct. Analysis, 1991]. This paper’s use of the Steinberg variety Zof triples reappeared in [72] in connection with the study of representations of anaffine Hecke algebra.

[39] (with D.Kazhdan) Schubert varieties and Poincare duality, 1980

The work on this paper was done in early 1979. The appendix to [37] showedthat the nontriviality of the polynomials Py,w of [37] (for ordinary Weyl groups)was very closely related to the failure of local Poincare duality on a Schubertvariety. It looked like the computations made in that appendix were actuallycomputations of local intersection cohomology in case of an isolated singularityor in the case where one meets the singular locus for the first time. (R.Bott hassuggested to Kazhdan that the results in that appendix could be related to inter-section cohomology. On the other hand I have attended a lecture of MacPhersonon intersection cohomology at Warwick in 1977 which dealt with the failure ofPoincare duality, as did the appendix to [37], and I was wondering about the con-nection between the two.) However, a preprint of Goresky, MacPherson gave adifferent value for the local intersection cohomology than what we found in ourcase. Therefore Kazhdan and I arranged to meet MacPherson (at Brown Univer-sity) to clarify this point. It turned out that the Goresky-MacPherson preprinthad a misprint and in fact it should have matched our computation. After thisKazhdan and I tried to identify all of Py,w with the local intersection cohomol-ogy of a Schubert variety and we succeeded in doing so (using results of Deligne).This is what is done in this paper. This can be viewed as a step in the proof ofthe conjecture on multiplicities in Verma modules in [37]. The idea to consider


the affine Schubert variety (attached to an element in the affine Weyl group) asan algebraic variety also appears (perhaps for the first time) in this paper. Thispaper also gives a proof of the positivity of coefficients of Py,w (for Weyl groupsand affine Weyl groups). In 2012 an elementary proof of the positivity valid forany Coxeter group was given by B.Elias and G.Williamson.

[40] Some problems in the representationtheory of finite Chevalley groups, 1980

This paper is based on a talk given in July 1979 at the Santa Cruz Conferenceon Finite Groups. It states several problems. Problem I states as a conjecture themultiplicity formula for unipotent representations in the virtual representationsR(T, 1) of [22]. This was solved in [42,45,46,57], (the last three papers make useof the results in [39]). Problem II is about assigning a unipotent support to anirreducible representation. This was solved in large characteristic in [100] and laterin general in [Geck, Malle, Trans. Amer. Math. Soc., 2000]. Problem III relatesthe families [34] of irreducible representations of the Weyl group with the two-sided cells [37]; it has been solved in [Barbasch and Vogan, Math. Ann. 1982 andJ.Alg. 1983]. Problem IV is a conjecture on the characters of irreducible modularrepresentations of a semisimple group in characteristic p > 0 in terms of thepolynomials of [37] attached to the affine Weyl group of the Langlands dual group.This was solved for p larger than a fixed unknown number by the combination of[Andersen, Jantzen and Soergel, Asterisque, 1994], [108,109,115,116], [Kashiwaraand Tanisaki, Duke Math.J, 1995, 1996], [117]. An explicit, rather large, boundfor p was found in [Fiebig, J. reine angew. math. 2012]. The bound found byFiebig cannot be much improved, see [Williamson, arxiv:1309.5055], based on [He,Williamson, arxiv: 1502.04914]. Problem V states that the unipotent classes of asemisimple group in large characteristic are in bijection with the two-sided cells(see [37]) of the affine Weyl group of the Langlands dual group. This was solvedin [86].

[41] Hecke algebras and Jantzen’sgeneric decomposition patterns, 1980

In this paper I introduce and study a certain module over an affine Heckealgebra, which I now call the periodic module. For simplicity I define it here intype A1. Let E be an affine euclidean space of dimension 1 with a given set P ofaffine hyperplanes (points) which is a single orbit of some nontrivial translation ofE. Then the group Ω of affine transformations of E generated by the reflectionswith respect to the various H in P is an infinite dihedral group. The connectedcomponents of E−P are called alcoves; they form a set X on which Ω acts simplytransitively. Let S be the set of orbits of Ω on P . It consists of two elements. Ifs ∈ S then s defines an involution A → sA of X where sA is the alcove 6= A suchthat A and sA contain in their closure a point in the orbit s. The maps A → sAgenerate a group of permutations of X which is a Coxeter group (W,S) (an affine

20 G. LUSZTIG

Weyl group of type A1 acting on the left on X). We assume that for any twoalcoves A,A′ whose closures contain exactly one common point (in P ) we havea rule which says which of the two alcove is to left (or to the right) of the otherin a manner consistent with translations. Let v be an indeterminate. Let H bethe Hecke algebra attached to W,S and let M be the free Z[v, v−1]-module withbasis X . There is a unique H-module structure on M such that for s ∈ S,A ∈ Xwe have TsA = sA if sA is to the right of A and TsA = v2sA + (v2 − 1)A if sAis to the left of A. For each H ∈ P let eH ∈ M be the sum of the two alcovesin X whose closures contain H. Let M0 be the H-submodule of M generatedby the elements eH . Now in the paper the higher dimensional analogue of thesituation above is studied. The analogue of X and the H-modules M,M0 areintroduced. A bar involution of M0 is introduced; it is semilinear with respectto the bar involution [37] on H. A canonical basis of the Z[v, v−1]-module M0

is constructed using the bar operator on M0 by a method similar to that of [37](but the construction is more intricate). This canonical basis is indexed by thealcoves in X. The polynomials which give the coefficient of an alcove B in the basiselement corresponding to an alcove A are periodic with respect to a simultaneoustranslation of A and B. They can be related to the polynomials attached in [37]to W; this relation proves a periodicity property for these last polynomials theproof of which was the main motivation for this paper. (An analogous periodicityproperty for the multiplicities in the Weyl modules of a simple algebraic groupin characteristic p was first pointed out by [Jantzen, J. Algebra, 1977] and theperiodicity result of this paper provided support for the conjecture in [40] on thesemultiplicities). Shortly after writing this paper I found the folowing geometricinterpretation of the results of this paper. Let G be a simply connected almostsimple group over C. Let K = C[[t]]. Let U be the unipotent radical of a Borelsubgroup of G and let I be an Iwahori subgroup of G(K). Then the set of doublecosets U(K)\G(K)/I is (noncanonically) the affine Weyl group and (canonically)the set X of alcoves as above. (A closely related statement is contained in [Bruhatand Tits, Groupes reductifs sur un corps local, Publ. IHES, 1972, Prop.(4.4.3)(1).)This led me to the statement that the periodic polynomials of this paper can beinterpreted as local intersection cohomologies of the (semiinfinite) U(K)-orbitson G(K)/I. This statement appears without proof in [59]; a proof appears in[Finkelberg and Mirkovic, Semiinfinite flags I; Feigin, Finkelberg, Kuznetsov andMirkovic, Semiinfinite flags II, Transl. of Amer. Math. Soc., 1999].

[42] On the unipotent characters of theexceptional groups over finite fields, 1980

In this paper I determine the multiplicities of the unipotent representations ofan exceptional group over a finite field Fq in the virtual representations R(w, 1)of [22], assuming that q is large. These multiplicity formulas were conjectured in[40]. The proof given in the paper uses the formulas (known at the time) for thedimensions of unipotent representations and unlike the later proof [54] (where the


restriction on q was removed) it does not use intersection cohomology methods.The method of this paper does not seem to be strong enough in the case of classicalgroups which was treated later in [45,46] using intersection cohomology methods.

[43] On a theorem of Benson and Curtis, 1981

Let H be the Hecke algebra over Q associated to the Weyl group W and to theparameter q, a power of a prime number. In 1964, Iwahori conjectured that H isisomorphic to the group algebra Q[W ]. Tits showed (in an exercise in Bourbaki)that this conjecture holds if Q is replaced by its algebraic closure. In 1972 Bensonand Curtis showed that Iwahori’s conjecture was true as originally stated butSpringer found a gap in the proof (for type E7). (The Benson-Curtis proof wascorrect for types other than E7, E8.) Springer in fact showed that the character ofa 512-dimensional irreducible representation of H (of type E7) definitely involves asquare root of q. In this paper (written in 1980) I construct an algebra isomorphismof Q[

√q]⊗H with Q[

√q][W ]. The key new observation is as follows. Consider the

vector space spanned by the elements of a fixed two-sided cell of W . There is a leftaction on this vector space for the Hecke algebra H with parameter q in which thebasis elements are identified with the elements of the new basis [37] of H; thereis also a right action on this vector space for the Hecke algebra H ′ with anotherparameter q′ in which the basis elements are identified with the elements of thenew basis [37] of H ′. The two actions obviously commute with each other if q = q′

but surprisingly they also commute with each other when q, q′ are independent.The proof is based on some properties of primitive ideals in an enveloping algebra.The isomorphism I construct is explicit unlike those in earlier approaches. Theuse of the theory of primitive ideals can nowadays be eliminated and replaced bythe use of the ”a-function” introduced in [60]. This paper also gives W -graphs (inthe sense of [37]) for the left cell representations of H in the noncrystallographiccase H3 and an example analogous to the 512-dimensional representation (for E7)is pointed out in type H3.

[44] Green polynumials and singularities of unipotent classes, 1981

In this paper I find a relation between:(1) the local intersection homology groups of the closure of a unipotent class in

GLn;(2) the local intersection homology of an affine Schubert variety in an affine

grassmannian of type A;(3) the character value at a unipotent element of an irreducible unipotent rep-

resentation of GLn(Fq).The connection (1)-(3) is a precursor of the theory of character sheaves which wasdevelopped in [63-65,68,69]. The connection (2)-(3) implies that the groups in (2)can be described in terms of multiplicities of weights for the finite dimensionalrepresentations of GLn(C) which was an inspiration for the paper [53] (a general-ization from GLn to a general reductive G). This paper also formulated the idea

22 G. LUSZTIG

(new at the time) that the Springer resolution is a small map and uses this ideato give a new definition of the Springer representations of a Weyl group in termsof intersection cohomology (unlike previous definitions this was valid in arbitrarycharacteristic). This shows in particular that the direct image of the constantsheaf under the Springer resolution is a perverse sheaf up to shift. Conjecture 2of this paper was subsequently proved by [Borho and MacPherson, C.R. Acad.Sci. Paris 1981]. The method introduced in this paper to construct Springer’srepresentations has been used in later papers:

(a) to construct the ”generalized Springer correspondence” [59];(b) to construct analogues of the Springer representation over parameter spaces

which yield representations of graded affine Hecke algebras [81];(c) to construct a version of Springer representations for affine Weyl groups

[125];(d) to construct a Weyl group action on the cohomology of certain quiver vari-

eties [149].

[45,46] Unipotent characters of symplectic and odd orthogonalgroups over a finite field, 1981; Unipotent characters of

the even orthogonal groups over a finite field, 1982

The first of these papers was conceived during a visit to the Australian NationalUniversity, Canberra (January, 1981); the second one was written later in 1981.Let G(Fq) be a group as in the title. In [30, Conj.4.3] I conjectured the precisepattern which gives the multiplicities of the various unipotent representations inthe virtual representations R(w, 1) of [22] or equivalently in the linear combinationsRE of the R(w, 1) with coefficients given by an irreducible character E of the Weylgroup; namely the pattern should be the same as the pattern [29] describing thedimensions of unipotent representations as linear combinations of fake degrees.This conjecture is established in this paper. The main new technique in the proofis the use of the local intersection cohomology of the closures of the varieties Xw

of [22] which I show that is the same as the local intersection cohomology of aSchubert variety and hence [39] is computable in terms of Hecke algebras. Anothernew technique used in the paper is the systematic use of the leading coefficientsof character values of the Hecke algebra. These techniques were later generalizedto any reductive group (see [57]).

[47] (with P.Deligne) Duality for representationsof a reductive group over a finite field, 1982

In 1977 I found a definition of an operation D in the complex representationgroup of a reductive group over Fq which to any representation E associates∑

P (sgnP )indP aa∗resP (E) where P runs over the parabolic subgroups over Fq

containing a fixed Borel subgroup over Fq , indP is induction from P (Fq) to G(Fq),a is lifting from P (Fq)/UP (Fq) to P (Fq), a

∗ is the adjoint of a and resP is restric-tion to P (Fq); sgnP is a sign. It was known at the time (Curtis) that D takes


the unit representation to the Steinberg representation. If E is cuspidal thena∗resP (E) is zero if P 6= G hence DE = ±E. But my main motivating examplewas one which I encountered in [13] where G = GLn(Fq), the complex numbers arereplaced by Fq and E is the natural representation of G on Fn

q . In that case DEcan be defined as above and can be viewed as a reduction mod p of a cuspidalcomplex representation of G of dimension (q − 1)(q2 − 1)...(qn−1 − 1); this wasthe main observation on which the work [13] was based. I conjectured that overcomplex numbers, D takes any irreducible E to an irreducible representation (upto sign) and that D2 = 1. In 1977 I communicated this conjecture to D.Alvis andC.W.Curtis (at the Corvallis Conference) and (separately) to N.Kawanaka. Myconjecture was proved (at the level of characters) by [Alvis, Bull. AMS, 1979],[Curtis, J. Algebra, 1980] and independently by [Kawanaka, Invent. Math., 1982].In the present paper a version of D at the level of representations (rather thancharacters) is given. As an application another proof of the conjecture is given.The operation D played a key role in my later work [57] where it was used toanalyze unipotent representations inductively (in conjunction with ”truncated in-duction” from a proper parabolic subgroup). An analogous operation plays a keyrole in the classification of character sheaves. In [A.M.Aubert, Trans.Amer. Math.Soc., 1995] a study of a p-adic analogue of the operation D defined in this paperis made.

[48] (with D.Alvis) On Springer’s representationsfor simple groups of type En (n = 6, 7, 8), 1982

Let G be as in the title (over C). In this paper we compute the Springerrepresentation of the Weyl group W of G corresponding to any unipotent classand the local system C on it. There are three tools that are used in the proof:(a) the compatibility of truncated induction with the Springer correspondence [35];(b) the conjecture (2) in [44] which was just proved by Borho and MacPherson; (c)an induction formula for the total Springer representation for a unipotent elementcontained in a proper Levi subgroup. Moreover, using the induce/restrict tables ofAlvis we showed that the class of irreducible representations of W thus obtainedcoincides with the class SW introduced in [36], thereby completing the proof ofthe conjecture at the end of [36] (which at the time of [36] was already known forclassical types and G2). In the appendix (by Spaltenstein) the rest of the Springercorrespondence (involving irreducible local systems 6= C) is determined.

[49] (with D. Alvis) The representations and genericdegrees of the Hecke algebra of type H4, 1982

In this paper the irreducible representatios of a Hecke algebra of type H4 areexplicitly constructed in terms of W -graphs. Moreover, the generic degrees ofthese representations are explicitly computed. Remarkably, these turn out to bepolynomials in q rather than rational functions. This fact suggested to me that atheory of unipotent representations for H4 should exist, and led to my paper [110].

24 G. LUSZTIG

[50] A class of irreducible representations of a Weyl group II, 1982

This paper was written in early 1981. Let W be a Weyl group and let IrrW bethe set of irreducible representations of W (up to isomorphism). In [34] a partitionof IrrW into subsets called families was described. The definition was such thatthe degrees of unipotent representations of a finite Chevalley groups were linearcombinations of fake degrees of objects of IrrW in a fixed family. In the presentpaper an elementary definition of families is given. More precisely a collection ofpossibly reducible representations (called cells and in later papers, constructiblerepresentations) is defined by induction. Namely it is required that by applying acertain kind of truncated induction to a cell of a proper parabolic subgroup oneobtains a cell of W ; moreover by tensoring a cell by the sign representation of Wone obtains again a cell. The cells are obtained by applying a succession of suchoperation starting with the trivial one dimensional representation of W . In thispaper the cells of any W are explicitly determined. It is shown that any E inIrrW appears in some cell; every cell contains a unique special representation (inthe sense of [36]) which in fact has multiplicity one; and two cells have a commonirreducible component if and only if they contain the same special representation.Therefore we can define an equivalence relation on IrrW as follows: E,E′ in IrrWare equivalent if there exist cells c, c′ such that E appears in c, E′ appears in c′ andc, c′ have the same special component. The equivalence classes are called families.In the paper it is conjectured that the cells of W are exactly the representationsof W that are carried by the left cells of W (in the sense of [37]). This conjecturewas proved in [70]. Using the results of this paper one can give a new definition ofthe involution of the set of special representations of W (see the comments to [36])which bypasses the consideration of a Galois group action: namely the involutionmaps a special representation E in a family f to the unique special representationin the family (f tensored by sign).

[51] (with D.Vogan) Singularities of closuresof K-orbits on a flag manifold, 1983

The work on this paper was done in late 1980. Its main object of study wasthe local intersection cohomology (l.i.c.) of the closure of a K-orbit on the flagmanifold of G where K is the identity component of the fixed point set of aninvolution of a complex reductive groupG. At the time it was known from the workof Beilinson and Bernstein that this l.i.c. is closely related to the computation ofmultiplicities in standard module of the various irreducible representations of a realreductive group attached to the involution in the same way as the l.i.c. of Schubertvarieties was known to be closely related to multiplicities in Verma modules. Theproblem of determining the l.i.c. in the present case was a generalization of theproblem of determining the l.i.c. of Schubert varieties solved in [39]. But themethod of [39] did not work in the present case, partly due to the presence ofnon-trivial equivariant local systems (of order two) on the K-orbits. Unlike in [39]


in this paper the connection with the representation theory of real groups is usedin the computation; also the purity theorem of Gabber (which was not availableat the time of [39]) plays a key role in the proof. The main result of this paperis that the l.i.c. are described in terms of some new polynomials Pγ,δ, where eachof γ and δ is a K-orbit together with a K-equivariant irreducible local system onit, which are explicitly computable and which generalize the polynomials Py,w of[37]. (Later work by Fokko Du Cloux has made possible the computation of Pγ,δ

on a computer.) This paper also contains an interpretation of the product in theHecke algebra and in certain modules over it in terms of convolution in derivedcategories (involving operations of inverse image, direct image and tensor productin derived categories). This interpretation which has become part of the folklorehas been also found around the same time by MacPherson, see [Springer, Sem.Bourbaki 589, 1982]. A proof of the results of this paper which is purely geometric(that is it does not rely on representation theory of real groups) has been laterfound by [Mars and Springer, Represent. Th., 1998].

[52] (with P.Deligne) Duality for representationsof a reductive group over a finite field, II, 1983

Let G be a connected reductive group over Fq. In this paper it is shown that the”duality operator” D of [47] applied to the virtual representation R(T, θ) in [22]is equal (up to sign) to R(T, θ). The proof is based on an inner product formulabetween R(T, θ) and an R(L, r) (as in [24]) where L is a Levi subgroup over Fq

of a parabolic (not necessarily over Fq) and r is a representation of L(Fq). Theproof of this orthogonality formula given in the paper contains a (not very serious)error. The corrected proof (which I supplied to Digne and Michel at their request)appears in the book [Digne, Michel, Representations of finite groups of Lie type,1991, 11.13].

[53] Singularities, character formulas anda q-analog of weight multiplicities, 1983

This paper was written in 1981 and presented at the Luminy Conference onAnalysis and Topology on Singular Spaces (July 1981). In this paper I find a veryclose connection between

-the category A of finite dimensional representations of a complex simply con-nected group G and

-the category A′ of G∗[[ǫ]]-equivariant perverse sheaves on the affine Grassman-nian associated to the Langlands dual G∗ of G.In more detail, let Λ+ be the set of dominant weights of G. For x ∈ Λ+, let Vx bethe finite dimensional irreducible representation of G corresponding to x and letmy(Vx) be the multiplicity of y ∈ Λ+ in Vx. Let Mx be the element of the (ex-tended) affine Weyl group W of G∗ which has maximal length in the double cosetof x with respect to the usual Weyl group W0. Let H be the affine Hecke algebra

26 G. LUSZTIG

of W and let (Cw)w∈W be the basis [37] of H. For x ∈ Λ+ we set γx = π−1CMx

where π = q−ν/2∑

w∈W0ql(w); here ν is the number of positive roots and l(w) is

the length of w. For x ∈ Λ+ let Ox be the closure of the G∗[[ǫ]]-orbit corespondingto x in the affine Grassmannian and let Πx be the corresponding simple object ofA′. For w,w′ in W let Pw′,w be the polynomial defined in [37]. Here are the mainresults of this paper.

(I) For x, y in Λ+ we have my(Vx) = PMy,Mx(1). (Thus the weight multiplicities

my(Vx) are related to the dimension of stalks of Πx.)

(II) For x, y in Λ+ we have γxγy =∑

z∈Λ+ cx,y,zγz where cx,y,z are naturalnumbers (apriori they are only polynomials in q). An equivalent statement is thatthe convolution Πx ∗Πy is a direct sum of objects Πz (z ∈ Λ+) without shifts; orthat the map which defines this convolution is semismall.

(III) For x, y, z in Λ+, the number cx,y,z in (II) is equal to the multiplicity ofVz in the tensor product Vx ⊗ Vy.

(IV) For x in Λ+, the vector space Vx is isomorphic to the total intersectioncohomology of Ox.(Statement (IV) appears in the last line of this paper; note that the odd intersec-tion cohomology of Ox is zero.)

Statement (II) is called the ”miraculous theorem” in [Beilinson and Drinfeld,Quantization of Hitchin integrable system... (1991), 5.3.6]. It is equivalent to thefact that A’ is a monoidal category under convolution. Statement (III) suggeststhat this monoidal category is equivalent to A with its obvious monoidal structureand statement (IV) suggests the definition of a fibre functor for A′ which wouldenter in the construction of such an equivalence. The tensor equivalence of A andA′ was established in [Ginzburg, arxiv:alg.geom./9511007] based on the results ofthis paper (using (II) and the fibre functor above), except that the commutativityisomorphism for A′ given there was incorrect and was later provided by Drinfeld(whose construction is sketched in [Mirkovic and Vilonen, Math. Res. Lett. 2000]).Thus the equivalence of A,A′ as monoidal categories (now known as the ”geometricSatake equivalence”) has been established by combining the ideas of this paperwith those of Ginzburg and Drinfeld. A version of the geometric Satake equivalencein positive characteristic is established in [Mirkovic and Vilonen, Math. Res. Lett.2000].

Now by (I) each weight multiplicity appears by setting q = 1 in a polynomialin q with positive coefficients; hence that polynomial can be viewed as a ”q-analogof weight multiplicities”, hence the title of the paper. Subsequently, a (partlyconjectural) interpretation of these q-analogs was given purely in terms of repre-sentations of G in [R.K.Gupta (later Brylinsky), Jour. Amer. Math. Soc. 1989];this was later confirmed in [Joseph, Letzter and Zelikson, Jour. Amer. Math. Soc.2000]. In this paper I also introduce a q-analogue of the Kostant partition functionand prove that it is equal to the q-analogue of weight multiplicities in the stablerange. The proof of (II) given in this paper relies on some dimension estimates in[41] involving semiinfinite geometry in disguise. Another result of this paper is a


description of the affine Grassmannian as an ind-variety (as a subset of the set ofselfdual orders in a simple Lie algebra over C((ǫ)) given by explicit equations).

We now change notation and assume that G is a simply connected algebraicgroup over an algebraically closed field of characteristic p > 0. Now Λ+ still makessense and for x ∈ Λ+, we denote by Lx the corresponding simple G-module.

One of my motivations to write this paper was to produce evidence for myconjecture on modular representations of G (Problem IV in [40]). More precisely,before writing this paper, I understood that the conjecture in [40] implies statement(I) above. Thus, a proof of (I) would be evidence for the validity of the conjecturein [40] and this was a motivation for me to try to prove (I). (At that time I alreadyknew that (I) is true in type A, as a consequence of [44].) I will now sketch how(I) can be proved assuming that the conjecture in [40] holds. From that conjecturewe have

ch(Lpx) =∑

y∈Λ+;y≤x

PMy,Mx(1)f−1

y (∑

w∈W0

sign(w)ch(Vpy+w(ρ)−ρ))

where fy is the order of the stabilizer of y in W0 and for any weight ν we set

ch(Vν) =∑

w′∈W0

sign(w′)ew′(ν+ρ)/

∑

w′∈W0

sign(w′)ew′ρ.

(We assume that p is large compared to x.) We have

∑

w∈W0

sign(w)ch(Vpy+w(ρ)−ρ)

=∑

w′,winW0

sign(ww′)epw′y+w′w(ρ)/

∑

w′∈W0

sign(w′)ew′(ρ) =

∑

w′∈W0

epw′y.

Thus,

ch(Lpx) =∑

y∈Λ+;y≤x

PMy,Mx(1)f−1

y

∑

w′∈W0

epw′y.

By the Steinberg’s tensor product theorem we have

ch(Lpx) =∑

y∈Λ+;y≤x

my(Vx)f−1y

∑

w′∈W0

epw′y.

We deduce that PMy,Mx(1) = my(Vx), as stated in (I).

[54] Some examples of square integrablerepresentations of semisimple p-adic groups, 1983

Let G be the group of rational points of a simple split adjoint algebraic groupover a nonarchimedean local field whose residue field has q elements. This paper

28 G. LUSZTIG

introduces the notion of unipotent representation of G; these are the irreducibleadmissible representations of G whose restriction to some parahoric subgroup ofG contain a unipotent cuspidal representation of the ”reductive” quotient of G.Let U ′ be the set of unipotent representations of G and let U be the subset of U ′

formed by the Iwahori-spherical representations of G.

According to the Deligne-Langlands conjecture, U is in finite to one corre-spondence with the set of pairs (s, u) where s, u are a semisimple element anda unipotent element (up to conjugacy) in the complex ”dual” group such thatsu = uqs.

One of the main contributions of this paper is the formulation of a refinementfor the Deligne-Langlands conjecture in which a third parameter is added to theDeligne-Langlands parameters, namely an irreducible representation ρ of the groupA(s, u) of connected components of the simultaneous centralizer of s, u on whichthe centre of the dual group acts trivially.

More precisely, in this paper I state the conjecture that the triples (s, u, ρ) asabove are in canonical bijection with U ′ and that U is in bijection with the set oftriples (s, u, ρ) such that ρ appears in the cohomology of the variety X of Borelsubgroups containing s and u.

The idea of this paper, to enrich a Langlands parameter by adding to it an irre-ducible representation of a certain finite group, has been also stated several yearslater (in more generality) in [Vogan, The local Langlands conjecture, Contemp.Math. 145, 1993].

A good thing about the refined conjecture (for U) is that it indicates that therepresentations of the affine Hecke algebra may be constructed geometrically interms of a space like X . The connection with geometry became even stronger afterthe equivariant K-theoretic approach of [66] was found and led to the solution ofthe (refined) conjecture for U in [67,72]. The (refined) conjecture for U ′ was estab-lished in [123]. I have arrived at the idea of the refined conjecture by experimentsperformed in this paper: I constructed explicitly (using W -graphs) the reflectionrepresentation and some closely related representations of the affine Hecke (and Ishowed that they are often square integrable by some very complicated computa-tion); these representations correspond conjecturally to the subregular unipotentelement and this provided evidence for the refined conjecture. In these examples Ialso found that the weight structure of the representations I construct can be in-terpreted in terms of the geometry of the varieties X above, further reinforcing theidea that the geometry of X should play a role in the proof of the conjecture. Inthis paper I introduce a description of the affine Weyl group of type A as a groupof periodic permutations of the integers. This point of view was later used exten-sively in [Shi, The Kahdan-Lusztig cells in certain affine..., Springer LNM 1179,1986]. I also give a conjecture giving the number of left cells in each two sided cellof an affine Weyl group of type A, which was later proved by [Shi, loc.cit.] and aconjecture describing explicitly the two sided cells of an affine Weyl group of typeA, which I later proved in [62]. Another result of this paper is the construction of


an imbedding of a Coxeter group of type H4 (resp. H3) into the Weyl group E8

(resp. D6) which has the property of sending any simple reflection to the productof two commuting simple reflections and any element of length n to an element oflength 2n (this is part of a general result about imbedding of Coxeter groups, see3.3.) This imbedding has been rediscovered ten years later in [Moody and Patera,J.of Physics,A, 1993].

The argument in 2.8 has been used in the later papers [67,72] to prove squareintegrability of certain geometrically defined representations of an affine Heckealgebra. The last sentence in 2.11 was later proved in [78].

[56] Open problems in algebraic groups, 1983

In the summer of 1983 I participated in a Taniguchi conference in Katata,Japan. The participants were asked to write up a list of open problems. Here aresome of the problems on my list.

(1) Let W be an affine Weyl group. Then the number of left cells contained inthe two-sided cell corresponding under the bijection in [86] to the conjugacy classof a unipotent element u in a reductive group over C of dual type to that of Wis equal to the dimension of the part ot the cohomology of the Springer fibre at uinvariant under the action of the centralizer of u.

(2) Let W be as in (1). We identify W with the set of (closed) alcoves in aneuclidean space in the standard way. Let A,B be two alcoves in the same twosided cell. Show that A,B are in the same left cell if and only if there exists asequence of alcoves A = A0, A1, ..., An = B (all in the same two sided cell) suchthat Ai, Ai+1 share a codimension 1 face for i = 0, 1, ..., n−1. Show that the unionof alcoves in a left cell is a contractible polyhedron. Show that similar results holdfor a finite Weyl group by replacing the euclidean space with the correspondingtriangulated sphere.

[57] Characters of reductive groups over a finite field, 1984

Let G be a connected reductive group with connected centre defined over Fq.The main contribution of this book (written in 1982) is the classification of theirreducible representations of G(Fq) and the computation of their multiplicitiesin the virtual representations R(w, θ) of [22]. (Earlier, this kind of results wereknown for unipotent representations with q large, see [42,45,46]; the classification(but not the multiplicities) for classical groups with any q was also known [29]).

Let L be a G-equivariant line bundle over the flag manifold of G and let L− 0be the complement of the zero section of L. Now Ch.1 contains

(∗) the computation of the local intersection cohomology of L − 0 with coef-ficients in certain ”monodromic” local systems on some smooth subvarieties ofL − 0, in terms of the polynomials [37] for the Weyl group of the centralizer of asemisimple element s in the dual group.

(This generalizes results of [39] which correspond to the case s = 1.) Sincethese local intersection cohomology groups were at the time known (by Beilinson

30 G. LUSZTIG

and Bernstein) to compute multiplicities in Verma modules with regular rationalhighest weight, (∗) was a new instance of a connection between representations of agroup and geometry of the dual group. Another proof of the multiplicity formulasin Verma modules with regular rational highest weight was later found in [Soergel,Jour. Amer. Math. Soc. 1990, Theorem 11] where these multiplicities are directlyrelated to analogous multiplicities for integral highest weight, thus bypassing (∗).Note that (∗) is used in [Beilinson and Bernstein, A proof of Jantzen’s conjecture,Adv. Sov. Math. 1993]. An affine generalization of (∗) is given in [117] where it isused as one of the steps in the proof of the character formula for quantum groupsof nonsimplylaced type at a root of 1. Finally, (∗) is used in Ch.2 of this bookto determine the local intersection cohomology of the closures of the varieties Xw

of [22] with coefficients in local systems associated with the covering Xw of Xw

described in [22]. Again the result is expressed in terms of the polynomials of [37]for the Weyl group of the centralizer of a semisimple element in the dual group.

[58] Characters of reductive groups over finite fields, 1984

This is based on my talk at the ICM-1982 held in Warsaw in 1983 (the 1982event was postponed due to the martial law). This paper is an exposition ofthe main results of [57] (written in 1982) which were under the assumption ofconnected centre. But in the present paper that assumption was removed. Inorder to remove two words: ”connected centre” from my paper I had to do twomonths of intensive work (June/July 1983) mainly with the case of Spin4n. Thesecomputations with spin groups (not included in the paper where no proofs weregiven) have been published 25 years later in [180] with some earlier hints given in[83].

[59] Intersection cohomology complexes on a reductive group, 1984

This paper was written in late 1982 and early 1983. Let G be a connectedreductive group over an algebraically closed field of characteristic p ≥ 0. Let X bethe (finite) set of all pairs (C,E) where C is a unipotent class in G and E is a G-equivariant irreducible local system on C (up to isomorphism). In the late 1970’sSpringer showed that (if p = 0 or if p is large) there is a natural bijection between acertain subsetX0 ofX and the set Irr(W ) of irreducible representations of the Weylgroup W of G. In [44] I gave a new definition of the Springer representations ofW using intersection cohomology methods which is valid without restriction on p,but the proof that it induces a bijection between X0 (which can be defined for anyp) and Irr(W ) was first given for arbitrary p in this paper, using a study of sheaveson the variety of semisimple classes. In this paper I show (extending the method of[44]) that a suitable enlargement of Irr(W ) is in canonical bijection (”generalizedSpringer correspondence”) with X itself. The enlargement is a disjoint union ofsets of the form Irr(Wi) where Wi is a collection of Weyl groups (one of which isW ). Of particular interest are the objects of X for which the corresponding Wi is1. These are the ”cuspidal local systems” (c.l.s.) which are introduced, studied


and classified in this paper. A G-equivariant local system E on a unipotent classC of G is a c.l.s. if for any proper parabolic P of G with unipotent radical UP andany unipotent g in P , the d-th cohomology with compact support of C∩gUP withcoefficients in E is zero (where d is dim(C) minus the dimension of the conjugacyclass of g in P/UP ); note that if d is replaced by d′ > d then the correspondingvanishing property holds for any E. A new feature of this paper is the explicitcombinatorial description of the generalized Springer correspondence in terms ofsome objects closely related to the ”symbols” in [29]. This was new even forthe ordinary Springer correspondence which was previously known only in theform of an algorithm (Shoji), rather than by a closed formula. In the case whereG is a spin group with p odd, the generalized Springer correspondence gives acombinatorial interpretation of the Jacobi triple product formula (see Section 14).Another new result of this paper was a definition of ”admissible complexes” on G,a class of perverse sheaves on G whose existence was conjectured in [57, 13.7,13.8]where the required class of perverse sheaves was defined for G = GLn. One of themain ingredients in the definition of admissible complexes is the notion of c.l.s.(see above) extended from unipotent classes to ”isolated classes”. The admissiblecomplexes on G reemerged in another incarnation (as ”character sheaves”) in theseries [63-65,68,69].

[60] Cells in affine Weyl groups, 1985

Let W be a Weyl group or an affine Weyl group. This paper develops sometechniques for computing the left/two-sided cells [37] of W . The main contributionof this paper is the definition of the function a : W −→ N. For w ∈ W , I define a(w)essentially as the order of the worst pole of the coefficient of Cw (the Hecke algebraelement of [37]) in a product TxTy of two (variable) standard basis elements of theHecke algebra. I show that a is constant on the two-sided cells of W . When W isof affine type the fact that a(w) is well defined needs a proof (given in the paper);in fact I show that a(w) is at most the number N of positive roots. Therefore theset W∗ = w ∈ W ; a(w) = N is of particular significance. Let W! be the set of allproducts abc where a, b, c ∈ W , the length of abc is the sum of the lengths of a, b, cand b has length N and is contained in a finite parabolic subgroup of W . In thepaper it is shown that W! ⊂ W∗; in particular, W∗ contains ”almost all” elementsof W . In this paper, using the function a, I describe explicitly the decompositionof the affine Weyl group W of type A2, B2, G2 into left/two-sided cells in termsof a picture in which W is viewed as the set of alcoves in a decomposition ofan euclidean plane and each alcove is colored according to the two-sided cell towhich it belongs. It turns out that, for affine A2, B2, G2 the number of two-sidedcells is 3, 4, 5; this was one of the pieces of evidence which led to my conjecture[40] (restated in this paper) on the relation between two-sided cells and unipotentclasses. From the results of this paper one can see that in rank 2 one has W∗ = W!

and W∗ is a single two-sided cell. This was extended to arbitrary rank in [Shi, J.Lond. Math. Soc. 1987] and [Bedard, Commun. in Alg. 1988].

32 G. LUSZTIG

[62] The two-sided cells of the affine Weyl group of type A, 1985

The results of this paper were presented at a conference at MSRI in May 1984.In early 1983 I have learned from R.Carter about the remarkable work of hisPh.D. student J.Y.Shi (at Warwick) in which Shi determined explicitly the leftcells of the affine Weyl group W of type An; it turned out that Shi’s methodswere not sufficient to determine the two-sided cells of W (for which I formulated aconjecture in [54]). After I introduced the function a on W in [60], I realized thatthe results of Shi together with the use the function a are sufficient to determinethe two-sided cells of W . This is what is done in this paper; see also [Shi, TheKazhdan-Lusztig cells in certain affine..., Springer LNM 1179, 1986].

[66] Equivariant K-theory andrepresentations of Hecke algebras, 1985

The work on this paper was done at the Tata Institute, Bombay, in December1983. At the time when this paper was written, the parameter q of a Heckealgebra was viewed as a number, an indeterminate, a Tate twist or a shift in aderived category. One of the main contributions of this paper is to formulate theidea (new at the time) to view q as the generator of the equivariant K-theory ofa point with respect to the circle group and that various modules of the affineHecke algebra H can be realized in terms of equivariant K-theory with respect toa group containing the circle group as a factor. More specifically in this paper Ishow that the principal series representations of H admits a description in terms ofequivariant K-theory as above and conjectures are formulated for a description inthe same spirit of other H-modules attached to nilpotent elements. The idea to useequivariant K-theory to study affine Hecke algebras was subsequently developed inthe papers [67, 72] (with Kazhdan) and in [Chriss and Ginzburg, Representationtheory and complex geometry, 1997]. The same idea was later used

-by Garland and Grojnowski (and by Varagnolo and Vasserot) to realize theCherednik (double affine Hecke) algebra;

-by Nakajima to realize geometrically an affine quantum group.

[73] Cells in affine Weyl groups, II, 1987

This paper is a continuation of [60]. Let W be a Weyl group or an affine Weylgroup. One of the main contributions of this paper is a definition (in terms ofthe function a of [60]) of a set D of involutions of W (which I call distinguishedinvolutions). The definition was inspired in part by a conjecture of [A. Joseph, J.Algebra 1981] for finite W , which in fact follows from the results of this paper.For finite W , one can identify D with the set of Duflo involutions defined in thetheory of primitive ideals; but I don’t know a similar identification for affine W .In this paper I show that each left cell contains exactly one element of D and thatthe set of left cells in W is finite (hence D is also finite). Note that the set of leftcells in a more general Coxeter group can be infinite, see [R.Bedard, Commun. in


Alg. 1986 and 1989]. The second main contribution of this paper is the definitionof the asymptotic Hecke ring J of W . This is a Z-module with basis tw;w ∈ Win which the multiplication constants are obtained from those for the new basis[37] of the Hecke algebra by making q tend to 0 (in a strange way, involving thea-function of [60]). It is not immediately clear that J is associative (it is so, due to[43]); this ring has a rather non-obvious unit element namely

∑

d∈D td. (Here thefiniteness of D is used). It is also shown that the Hecke algebra admits a naturalalgebra homomorphism into the algebra J with scalars suitably extended (this isagain based on [43]).

[78] (with C. DeConcini and C. Procesi) Homology of the zeroset of a nilpotent vector field on a flag manifold, 1988

The work on this paper was done during my sabbatical leave in Rome (1985/86).Let g be the Lie algebra of a connected reductive group overC, letN be a nilpotentelement of g and let BN be the variety of Borel subalgebras of g that contain N . Atthe time this paper was written it was known that the rational homology of BN iszero in odd degrees. (The most difficult case, that of type E8, was done by [Beynonand Spaltenstein, J. Algebra 1984] based on computer calculation and then inmy paper [69] without computer calculation.) In this paper we prove a strongerresult namely that the integral homology of BN is zero in odd degrees and has notorsion in even degrees. The key case is that where N is distinguished. There areseparate proofs for the case of classical groups (where we show the existence ofa cell decomposition) and in the exceptional case (where we are unable to provethe existence of a cell decomposition but instead we give an alternative argumentbased on blow ups and downs which in a sense gives a more precise result thanfor the classical groups). It would be interesting to complete the results of thispaper by 1) extending the method used for exceptional groups (connectedness of acertain graph) to classical groups and 2) showing that the cell decomposition alsoexists for exceptional groups. In this paper we also show that the Chow group ofBN is the same as the integral homology. This has the consequence that the K-theory of coherent sheaves on BN is computable, which is a necessary ingredientof [140,143] and also of [Bezrukavnikov and Mirkovic, arxiv:1001.2562].

[79] Quantum deformations of certain simplemodules over enveloping algebras, 1988

In 1986, A. Borel wrote to me a letter pointing out the interesting new work ofJimbo in which quantized enveloping algebras (q.e.a) were introduced. As a resultof this letter I gave a course (1986/87) at MIT on q.e.a. and this paper came out

of it. In this paper the divided powers E(n)i , F

(n)i are introduced for the first time

by replacing the denominator n! of the classical divided powers by a q-analogueof n! (depending on i). The choice of denominator was such that the formulas for

the action of E(n)i , F

(n)i on the standard simple modules of quantum sl2 were as

34 G. LUSZTIG

simple as possible and also the quantum Serre relations can be written in a formwhich is as simple as possible. Using these divided powers, in this paper I define aQ[q, q−1]-form of the q.e.a. (In later papers [90,91] this was refined to a Z[q, q−1]-form which has become one of the ingredients in the definition of the canonicalbasis [92].) Using this I show that a simple integrable module of a Kac-MoodyLie algebra can be deformed to a module over the corresponding q.e.a. This paperalso contains the first appearance of the braid group action on a q.e.a. at least inthe simply laced case (but the proofs appeared only in [107]).

[80] (with D.Kazhdan) Fixed points on affine flag manifolds, 1988

My motivation for this paper was as follows. Let G be a semisimple adjointgroup over C with Lie algebra g. Since [40] I knew that (conjecturally) the nilpo-tent classes of g are in bijection with the two-sided cells of the affine Weyl groupWaf of G∗ (Langlands dual); moreover experiments showed that dimH∗(Bx)

A(x)

(where Bx is the Springer fibre at a nilpotent x and A(x) is the =group ofcomponents of the centralizer of x in G) is equal to the number of left cells inthe corresponding two-sided cell. For example if G is of type E8 and x is asubregular nilpotent element, then Bx has 8 irreducible components (all lines),H∗(Bx)

A(x) = H∗(Bx) is 9-dimensional and there are 9 left cells in the corre-sponding two-sided cell c. If we now take the affine analogue x′ ∈ g((t)) of asubregular nilpotent x in g and if we replace Bx by the set B′

x′ of Iwahori subalge-bras of g((t)) that contain x′ we see that B′

x′ has exactly 9 irreducible components(all lines). So the number of left cells in c can now be interpreted not as a di-mension of a vector space but as a number of elements in a set attached to x′

(the set of irreducible components of B′x′). This gave me some hope of finding an

analogous relation in more general cases. Although this hope remained unfulfilledit motivated my interest in investigating sets of the form B′

x′ . In this paper it isshown that if N ∈ g((t)) is, like x′ above, regular semisimple and topologicallynilpotent (that is limNk = 0 as k → ∞) then B′

N (defined as for x′) is a finiteor countable (but locally finite) union of projective algebraic varieties all of thesame dimension; moreover if N is in addition elliptic then B′

N is itself an algebraicvariety. In the paper a conjectural formula for dim(B′

N ) is given and it is shownhow to reduce the proof of this formula to the case where N is elliptic. (The casewhere N is elliptic was settled in [R.Bezrukavnikov, Math. Res. Lett. 1996].)Also, it is shown that if x ∈ g is nilpotent then for an ”open dense” subset S(x)of x + tg[[t]], all elements N ∈ S(x) are regular semisimple (and of course topo-logically nilpotent), dimB′

N = dimBx and the conjugacy class in the Weyl groupwhich parametrizes the Cartan subalgebra of g((t)) containing N depends only onx. (For example if x ∈ g is subregular nilpotent then x′ can be taken to be anelement of S(x); note that x → N is an affine analogue of the process of induc-tion [35]. This gives a map Ψ from nilpotent orbits of g to the set of conjugacyclasses in the Weyl group. In the paper this map is described explicitly in typeA (where it is a bijection) and in the cases arising from a nilpotent element of


g whose centralizer in G is connected, unipotent. For example if G has type E8

and x is regular/subregular/subsubregular then Ψ(x) contains an element of order30/24/20. The map Ψ was later computed for G of type B,C,D in [N.Spaltenstein,Asterisque 168(1988)], [N.Spaltenstein, Arch. Math. (Basel) 1990], and in manycases in exceptional types in [N.Spaltenstein, Adv. Math. 1990]. But there areseveral cases in exceptional groups where Ψ(x) remains uncomputed. In [207]another map between the same two sets is defined using completely different con-siderations (based on [199] where a map in the opposite direction is defined usingproperties of Bruhat decomposition). The map in [207] is computable in all casesand I expect it to be the same as Ψ. The varieties B′

N introduced in this paperplay a key role in the work of Ngo B.C. on the fundamental lemma.

I would like to state the following problem. Let x ∈ g be a distinguished nilpo-tent element and let N ∈ S(x) (so that B′

N is a well defined algebraic varietycontaining Bx, see Cor.2 in Sec.3 and 9.2). Let XN be the set of irreducible com-ponents of B′

N . Show that A(x) acts naturally on XN , that dimH∗(Bx)A(x) =

card(XN/A(x)) and that the set of left cells in the two-sided cell of Waf corre-sponding to x is in natural bijection with XN/A(x). For example if G is of type G2

(resp. F4) and x is subregular then A(x) = S3 (resp. S2) and dimH∗(Bx)A(x) = 3

(resp. 5); on the other hand, B′N is a Dynkin curve of type affine E6 (resp. affine

E7) which has a natural S3-action (resp. S2-action) whose fixed point set on theset of irreducible components has cardinal 3 (resp. 5).

[81] Cuspidal local systems and graded Hecke algebras, I, 1988

The work on this paper was done in late 1987. In this paper I introduce a gradedanalogue of affine Hecke algebras (with possibly unequal parameters) associatedto any root system. (After this paper was written I learned of the paper [Drinfeld,Funkt. Anal. Appl. 1986] where a similar algebra was introduced for a root systemof type A and with the grading being disregarded.) Another new idea of thispaper is to define equivariant homology. While in Borel’s definition of equivariantcohomology with respect to an action of an algebraic group G, any classifying spaceof G can be used, the definition that I give for equivariant homology is more subtle:it exploits the fact that the classifying space of G can be approximated by smoothvarieties. (The same idea appeared independently in the definition of equivariantderived category given in [Bernstein and Lunts, LNM, Springer Verlag 1994].)This paper contains also a new application of the theory of character sheaves.Originally this theory was supposed to provide a machine to compute the valuesof the irreducible characters of a reductive group over a finite field. But in thispaper character sheaves (cuspidal with unipotent support) are used to constructgeometrically representations of a graded Hecke algebra (which ultimately leads torepresentations of a p-adic group [123,155]). In fact, in this paper I give a geometricrealization of certain graded Hecke algebras in terms of equivariant homology ofa space with group action and with a local system associated to a cuspidal localsystem with unipotent support.

36 G. LUSZTIG

[84] Modular representations and quantum groups, 1989

The work on this paper was done in the spring of 1987 and the results werepresented at a US-China conference at Tsinghua University, Beijing in the summerof 1987. This paper introduces a new concept: that of the quantum group Uζ (ζ is a

primitivem-th root of 1 in the complex numbers) obtained from theQ[q, q−1]]-formof the quantum group introduced in [79] (which involves q-analogues of dividedpowers) by specializing q = ζ. This paper also formulates the idea (new at thetime) that, in the case where m is a prime number p, the representation theoryof Uζ is governed by laws similar to those of the rational representation theoryof a semisimple algebraic group G over a field of characteristic p. Most of thepaper is concerned with providing evidence for this idea. For example, I provean analogue for Uζ of the Steinberg tensor product theorem [Steinberg, NagoyaJ.Math. 1963]. Thus I show that a simple module of Uζ with highest weightλ = λ0 + pλ1 (where λ0 has coordinates strictly less than p) is the tensor productof the simple module of Uζ with heighest weight λ0 with a Uζ-module which may beviewed as the simple U1-module with highest weight λ1. (Implicit in this statementis the existence of a ”quantum Frobenius homomorphism” from Uζ to the classicalenveloping algebra U1 which is also one of the main new observations of this paper.)The key to this tensor product theorem is the following property of the Gaussianbinomial coefficients specialized at ζ: if N,R are integers and N = N0 + pN1, R =R0 + pR1 where Ni, Ri are integers, 0 ≤ N0 ≤ p − 1, 0 ≤ R0 ≤ p − 1, then[N,R] = [N0, R0](N1, R1) for q = ζ. Here [N,R], [N0, R0] are Gaussian binomialcoefficients and (N1, R1) is an ordinary binomial coefficient. Similarly, the keyto the classical Steinberg theorem is the following congruence (which I learnedin my student days from Steenrod’s book ”Cohomology operations”): if N,R areintegers and N = N0+pN1+p2N2+ ..., R = R0+pR1+p2R2+ ... where Ni, Ri areintegers, 0 ≤ Ni ≤ p−1, 0 ≤ Ri ≤ p−1, then (N,R) = (N0, R0)(N1, R1)(N2, R2)...mod p. In this paper I also formulate a conjecture describing the character of anirreducible finite dimensional Uζ-module in terms of the polynomials [37] attachedto the affine Weyl group of the Langlands dual group, similar to the conjecturethat I stated in [40, Problem IV]. This conjecture (which is now known to hold)is one of the steps in the solution of Problem IV in [40].

[86] Cells in affine Weyl groups IV, 1989

One of the main results of this paper is establishing a bijection between theset of unipotent classes in a connected reductive group G over C and the setof two-sided cells in the (extended) affine Weyl W group associated to the dualgroup G∗. (This was conjectured in [40].) The proof uses the earlier parts ofthis series (especially the study of the J-ring associate to W ) and the results ofof [72]. Assume now that G is simply connected. One of the main contributionsof this paper is the formulation (see 10.5) of a (conjectural) basis preserving ringisomorphism between the J-ring ofW and the direct sum over the unipotent classes


of G of certain equivariantK-groups of certain finite sets attached to the unipotentclasses. I arrived at this conjecture after doing many explicit computations for rank2 and using an analogy with finite Weyl groups [77].

A weaker form of this conjecture is proved in [Bezrukavnikov, Ostrik, in Adv.Studies Pure Math.40 Mat. Soc. Japan 2004]; for type A the conjecture is provedin full in [Xi, Mem. Amer. Math. Soc. 157(2002)]. A consequence of theconjecture (see 10.8) gives a conjectural bijection between the set of dominantweights of G and the set of pairs consisting of a unipotent class of G and anirreducible rational representation of the centralizer of an element in that class.This has now been established in [Bezrukavnikov, Represent. Th. 2003] with animportant contribution by [Ostrik, Represent. Th. 2000].

[88]. On quantum groups, 1990

This paper (written in early 1989) consists of two parts. In the first part itis shown that from a quantum group associated to a positive definite symmetricCartan matrix one can recover in a natural way the Hecke algebra attached tothe same Cartan matrix. Namely, an explicit construction of the q-analog of theadjoint representation is given (together with an explicit basis which can now beinterpreted as the canonical basis [92] of that representation) and it is shown thatthe braid group acts naturally on this representation so that the induced actionon the 0-weight space satisfies the relations of the Hecke algebra. In the secondpart two conjectures are formulated. Conjecture 2.3 predicts an equivalence ofcategories between a certain category C of representations of a quantum group ata root of 1 and a certain category C′ of representations of an affine Lie algebra ata negative central charge related to the order of the root of 1. (This conjecturewas later proved in [108,109,115,116].) Conjecture 2.5(b) (resp.2.5(c)) predicts acharacter formula for the simple objects in C′ (resp. C) in terms of the polynomials[37] for an affine Weyl group analogous to a conjecture I made in [40] for modularrepresentations of a semisimple group in characteristic p. Conjecture 2.5(b) hasbeen already stated in [84] but the present paper suggested that one could prove itif one could prove Conjectures 2.3 and 2.5(c). Eventually that was indeed the waythat Conjecture 2.5(b) was proved. (Conjecture 2.5(c) was proved by [Kashiwaraand Tanisaki, Duke Math.J. 1995]).

[90] Finite dimensional Hopf algebras arizing fromquantized universal enveloping algebras, 1990

This paper was written in the spring of 1989. Let A = Z[v, v−1]. This paperintroduces a new object: the A-form AU and AU

+ of a quantized envelopingalgebra U of simplylaced type and its plus part U+. While the definition doesnot need new ideas (compared to the definition of the Q[v, v−1]-form of U or U+,already introduced in [79]) the problem that arises is to show that one gets awell behaved object, for example that AU

+ is a ”lattice” in U+. This property isestablished in the present paper by constructing an A-basis for AU

+ which is also

38 G. LUSZTIG

a basis of U+. This basis is defined in terms of the braid group action introducedin [79]. The proof relies on explicit calculations involving (in particular) the rootsof E8. In this paper I also introduce for an integer N > 0 a new Hopf algebra offinite dimension Nnumber of roots which can be viewed as the Hopf algebra kernelof the quantum Frobenius map of [84]. This Hopf algebra is sometimes referredto as the ”small quantum group”. This paper is a step toward the construction ofthe canonical basis of U+ which was achieved in [92]. Indeed, the lattice AU

+ isone of the key ingredients in the definition of the canonical basis of U+ given in[92].

[89] Green functions and character sheaves, 1990

I got the main idea for this paper during a visit at the College de France(May 1988) where I gave a series of lectures on character sheaves. The paper wascompleted in the fall of 1988 when I was visiting IAS, Princeton. This paper is astep in the program (initiated in [64, p.226]) of relating (for a connected reductivegroup G defined over Fq of characteristic p), the characters of representationsof G(Fq) and the characteristic functions of character sheaves on G which are”defined” over Fq. A part of this program would be to show that the Greenfunctions of G(Fq) (defined in [22]) can be expressed in terms of character sheaves.In this paper I show that this is indeed so assuming that q is large (no restrictionon p). The corresponding result for large p was known at the time (it could bededuced from the work of Springer and Kazhdan). The assumption that q is largeenough was later removed by [Shoji, Adv.in Math.1995]. Moreover, in this paperit is shown that the ”generalized Green functions” associated to the ”induction”functor RG

L,P of [24] can be expressed in terms of character sheaves assuming thatq is large enough and p is good. This was new even for large p. In fact theassumption that p is good can now be removed in view of the cleanness property[204]. The methods and results in this paper were used in [Shoji, Adv.in Math.1995 and 1996] to study my conjecture [64, p.226] on the relation of irreduciblecharacters of G(Fq) and character sheaves.

[91]. Quantum groups at roots of 1, 1990

In this paper the definition of the braid group action in [79], the results of[90] about the Z[v, v−1]-form of U+ (”lattice property”) and the definition of thesmall quantum group in [90] are extended to the nonsimplylaced case. The case ofG2 was particularly complicated since (unlike the other rank two cases) there areno simple explicit formulas for the commutation of two divided powers of ”rootvectors” and for this reason the argument becomes involved. Also the quantumFrobenius homomorphism (which is almost explicit in [84]) is made explicit. Inthe Appendix (joint work with M. Dyer) the ”Poincare-Birkhoff-Witt basis” ofU+ corresponding to any reduced expression of the longest Weyl group elementis introduced, using the braid group action and the computations in rank 2 fromthe main body of the paper. Note that the basis introduced in [89] is a special


case of this PBW basis; the appendix allows one to simplify some arguments in[89]. Later, these PBW bases turned out to be another of the key ingredients inthe definition of the canonical basis of U+ (in the simplylaced case) given in [92].

[92] Canonical bases arising fromquantized enveloping algebras, 1990

The results of this paper were obtained while I was giving a course (MIT, fall1989) on quantum groups and in particular on Ringel’s work [Ringel, Hall algebrasand quantum groups, Inv. Math. 1990] and were presented in that course. Thispaper introduces a rather miraculous object: the canonical basis for U+, the pluspart of a quantized enveloping algebra U of type A,D,E. In the paper this is doneby two methods (which lead to the same basis):

(1) an algebraic one based on the following three ingredients: (i) an integerform of U+ which I introduced earlier [79,90], (ii) a bar involution of U+ and (iii)a basis at infinity of U+ coming from any PBW basis, see [91] (remarkably, thebasis at infinity defined by a PBW basis is independent of the PBW basis);

(2) a topological method based on the local intersection cohomology of the orbitclosures in the moduli space of representations of a quiver.Now even in the approach (1), there is a (minimal) use of the elementary represen-tation theory of quivers (not intersection cohomology and not in the statementsbut in the proofs). Note that (1) (resp. (2)) bear some superficial similarity withthings which appeared in the study of Hecke algebras [37] (resp. [39]); in thatstudy the role of PBW bases is played by the (single) standard basis of the Heckealgebra. One of the remarkable properties of the canonical basis is that it inducesa basis in each finite dimensional irreducible module of U . This paper introducesalso a natural piecewise linear structure for the canonical basis that is, a finitecollections of bijections of the canonical basis with Nn (n=number of positiveroots) so that any two of these bijections differ by composition with a bijection ofNn with itself given by a composition of operations which involve only the sum ordifference of two numbers or the minimum of two numbers. Later, I found that ex-actly the same pattern appears in a rather different context: the parametrizationof the totally positive semigroup attached to a group of type A,D,E, see [119].A similar pattern exists in the nonsimplylaced case, see [193]. Theorem 8.13 giveswhat is I believe the first purely combinatorial formula for the dimension of a fi-nite dimensional irreducible representation (and its weight spaces) of a simple Liealgebra of type A,D,E (it expresses the dimension as the result of counting thenumber of elements of an explicit set defined using the piecewise linear structureabove); the previously known dimension formulas gave the dimension as a ratioof two integers which is not obviously an integer (Weyl) or as a difference of twointegers which is not obviously positive (Kostant). Subsequently, another purelycombinatorial formula was found by Littelman using his paths. The remarks onFourier transform in Sec.13 are a precursor of [97].

Another proof of the existence of the canonical basis (valid also for Kac-Moody

40 G. LUSZTIG

Lie algebras) was later given in [Kashiwara, Duke Math.J. 1991] by a purely al-gebraic method which again uses (i),(ii) above; and in [97] which generalizes theintersection cohomology approach (2).

The transition matrix between the canonical basis of U+ and a fixed PBW ba-sis of U+ attached to a reduced expression of the longest Weyl group element hasentries which are positive. This is proved in the paper for certain special reducedexpressions (adapted to an orientation of the Coxeter graph) when these entriesare interpreted as local intersection cohomology of orbit closures. The same state-ment for an arbitrary reduced expression is proved in [Syu Kato, arxiv:1203.5254].Another proof (relying on the positivity property of the comultiplication provedin [97]) appears in [H.Oya, arxiv:1501.01416.]

[95] Canonical bases arising fromquantized enveloping algebras, II, 1990

Some time after my paper [92] became available, Kashiwara found a differentapproach to the canonical basis of [92] in which he preserved two of the ingredientsin my definition ((1) it is contained in the “integral part” of the quantum groupand (2) is fixed by a bar operator) but replaced ny third ingredient with a differentone which made sense in the more general context of Kac-Moody type). One of theresults of this paper was that for ADE types, Kashiwara’s definition gives the sameresult as my original definition. Another contribution of this paper is the definitionof a new variety attached to an arbitrary graph. It is shown that these varietiesare equidimensional. (Each one is in fact a Lagrangian variety in a symplecticvector space.) Moreover the union Z of the sets of irreducible components of thesevarieties is endowed with certain geometrically defined maps Ek : Z → Z (see8.8); k is a vertex of the graph. In this paper it is conjectured (10.2) that thecrystal graph of the plus part of the quantized enveloping algebra correspondingto the graph can be geometrically realized as the set Z together with our mapsEk : Z → Z (there are also maps Fk : Z → Z but they are essentially inverse toEk hence they need not be separately constructed). This conjecture was provedby [Kashiwara and Y.Saito, Duke Math.J. 1997].

[97] Quivers, perverse sheaves andquantized enveloping algebras, 1991

Let U+ be the plus part of the quantized enveloping algebra correspondingto a symmetric Cartan matrix C. After writing the paper [92] on the canonicalbasis of U+ in the case where C is positive definite, I tried to consider the similarproblem for a general C. The main problem was to find an appropriate definitionfor the class X of irreducible perverse sheaves on the space of representations offixed dimension D of a quiver attached to C which should constitute the canonicalbasis. If C is positive definite, X consists of all G-equivariant simple perversesheaves (G=product of GLn’s); but in the indefinite case there are infinitely manyG-equivariant simple perverse sheaves which is not what X should be. I first tried


[95] to define X by imposing in addition to G-equivariance a condition on thesingular support namely that it should be contained in the explicit Lagrangianvariety Λ defined in [95]. But I was not able to develop the theory from thisdefinition. Instead I adopted a definition from the theory of character sheaves,namely X is defined as the collection of simple perverse sheaves which appear (upto shift) as direct summands of the direct image of the constant sheaf under theprojection maps from certain spaces which consists of a representation of dimensionD of the quiver and a ”flag” of a fixed type compatible with the representation.This makes X finite for any prescribed D. With this definition the collection ofthe various X when D varies can be viewed as a basis of an algebra over Z[q, q−1]in which multiplication is an analogue of induction of character sheaves (q appearsas the shift). In this paper I prove that the resulting algebra is a Z[q, q−1]-formof U+ and that the basis of U+ provided by the perverse sheaves does not dependon the orientation of the quiver; hence it is a canonical basis of U+. I also showthat this algebra has something close to a comultiplication (it is defined as ananalogue of restriction of character sheaves). The structure constants of boththe multiplication and ”comultiplication” are in N[q, q−1]. Another result of thispaper is a new realization of the algebra U+ (for v = 1) in terms of convolutionof certain constructible functions on the Lagrangian variety Λ (as above). Thisrealization actually plays a role in the proofs in this paper.

[98]. (with J.M.Smelt) Fixed pointvarieties in the space of lattices, 1991

Let V be a vector space of dimension n over C[[ǫ]] with a basis e1, ..., en. LetI be the space of Iwahori subalgebras of SL(V ) (an affine flag manifold). LetN be the linear map from V −→ V such that N(ei) = ei+1 for i = 1, ..., n − 1,N(en) = ǫe1. Let t > 0 be an integer relatively prime to n. In this paper westudy the space Xt = B ∈ I;N t ∈ B (by [80], Xt is a projective algebraicvariety over C). It is shown that the Euler characteristic of Xt is χ(Xt) = tn−1

and that Xt can be paved with affine spaces. After this paper appeared, I defineda generalization of N t for any simple Lie algebra g over C; namely for an integert ≥ 1 prime to the Coxeter number h we write t = ah + b, 1 ≤ b ≤ h − 1and let Nt = ǫa

∑

α: root cαeα where eα are the root vectors and cα = 1 if theheight of α is b, cα = ǫ if the height of α is h − b, cα = 0 if the height of αis not b or h − b; then Nt is a topologically nilpotent regular semisimple ellipticelement of Coxeter type. Let Xt be the variety of Iwahori subalgebras of g[[ǫ]] thatcontain Nt (a projective variety); I conjectured that the Euler characteristic of Xt

is χ(Xt) = trank(g), which in type A reduces to the formula in this paper. Thisconjecture was proved in [Fan, Transfor. Groups, 1996]. The result on paving wasgeneralized in [Goresky, Kottwitz and MacPherson, Represent. Th.,2006]. In thispaper there is also an explicit formula for the Euler characteristic in the case wherethe space of Iwahori subalgebras is replaced by that of maximal parahoric algebras(type A); this was generalized to arbitrary g in [Sommers, Nilpotent orbits and

42 G. LUSZTIG

...(Ph.D.Thesis at MIT), 1997]. The formula for χ(Xt) in this paper plays a rolein [Berest, Etingof and Ginzburg, IMRN, 2003]. The variety Xt (type A) and itspaving in this paper also plays a role in [Laumon, Fibres de Springer et jacobiennescompactifiees, Springer 2006].

[100] A unipotent support for irreducible representations, 1992

Let G be a connected reductive group defined over a finite field Fq of sufficientlylarge characteristic. For any unipotent element u ∈ G(Fq) let Γu be the generalizedGelfand-Graev representation (GGGR) associated by Kawanaka to u; this is arepresentation of G(Fq) whose character is zero outside the unipotent set. Let ρbe an irreducible complex representation of G(Fq); let ρ

′ be the representation ofG(Fq) which is dual to ρ in the sense of [47]. In [57, 13.4] a unipotent conjugacyclass C of G was attached to ρ. In this paper the following properties of C areproved (see Theorem 11.2).

(i) The average value of the character of ρ on C(Fq) is nonzero and C is charac-terized by having maximum dimension among unipotent classes with this property.

(ii) If g ∈ G is such that tr(g, ρ) 6= 0 then the unipotent part of g lies in C orin a conjugacy class of dimension < dimC.

(iii) For some u ∈ C(Fq), ρ′ appears with non-zero multiplicity in Γu; for any

u ∈ C(Fq), ρ′ appears with small multiplicity in Γu; if C

′ is a unipotent class inG such that dim(C′) > dim(C) or dim(C′) = dim(C), C′ 6= C, then ρ′ does notappear in Γu for u ∈ C′(Fq).Note that something close to (i) has been conjectured in [40]; (ii) has been hintedat in [76,p.177,line 13]; (iii) has been conjectured by Kawanaka. It is natural to callC the unipotent support of ρ. One of the keys to the proof of (i)-(iii) is Theorem7.3 of this paper which gives an explicit decomposition of a GGGR in terms ofintersection cohomology complexes of closures of unipotent classes with coefficientsin various local systems. A step in the proof of this theorem is a formula for theFourier transform of a GGGR viewed as a function on Lie(G(Fq)), involving aSlodowy slice. The connection between GGGR and Slodowy slices found in thispaper is perhaps related to the observation made several years later by [Premet,Special transversal slices ..., Adv.in Math.2002] that a W -algebra (a characteristiczero analogue of the endomorphism algebra of a GGGR) is a quantized version ofthe coordinate ring of a Slodowy slice. In this paper we also give a (provisional)definition (see Theorem 10.7) of the unipotent support of a character sheaf on G.The actual definition (partly conjectural) is given in [212].

[104] Affine quivers and canonical bases, 1992

In this paper I fix an affine quiver of type A,D or E (but not A2n) with oneof the two orientations in which every vertex is a sink or a source. In this case Iconstruct explicitly the perverse sheaves on the space of representations of fixeddimension of the quiver which constitute the canonical basis introduced in [97].


Unlike in the finite type case, these perverse sheaves can be higher dimensionallocal systems on an open subset of their support (the dimension is that of anirreducible representation of a symmetric group). Also, I describe explicitly (enu-merate) the irreducible components of the Lagrangian variety Λ attached in [95]to the affine quiver and show that they are in natural bijection with the perversesheaves in the canonical basis. In this paper, the affine quivers are studied interms of a finite subgroup of SL2(C) (MacKay correspondence) and I reprovefrom this point of view the classification of the indecomposable representations ofthis quiver, which goes back in various degrees of generality to Weierstrass andKronecker (affine A1), Gelfand and Ponomarev (affine D4), Donovan and Freislich,Nazarova and [Dlab and Ringel, Memoirs AMS, 1976]. Another result of this pa-per is the construction of a new basis of the algebra U+ (with v = 1) attached toour quiver (later called the semicanonical basis [147]) in which the basis elementsappear as constructible functions on the Lagrangian variety Λ.

[110] Coxeter groups and unipotent representations, 1993

This paper contains things that I did in 1982. One of the results of the classifi-cation [57] of unipotent representations of a Chevalley group over Fq was that theset of unipotent representations depends only on the Weyl group W , not on theunderlying root system or Chevalley group. Therefore one can asks whether theset of unipotent representations makes sense if W is replaced by a finite Coxetergroup when the root system and the Chevalley group are not defined. (One indi-cation that this may be true was provided by the results of [49] which computedwhat should be the degrees of the principal series of unipotent representations intype H4 and these degrees turned out to be polynomials in q.) This question isanswered in this paper: the set of unipotent representations is attached to the fi-nite Coxeter group W by heuristic considerations by postulating certain propertiesthat this set should have which are known in the crystallographic case and showingthat these postulates have a unique solution in the general case. The degrees ofthe unipotent representations are computed (extending the results of [49]) and theclassification of representations in families, the classification of unipotent cuspidalrepresentations are given in each noncrystallographic case. For example if W isof type H4 there are 104 unipotent representations of which 50 are cuspidal; thelargest family contains 74 representations of degree cq6+higher powers of q wherec is an algebraic integer (independent of q) divided by 120. This result has beenfound independently by Broue and Malle (unpublished). It has become a part ofa heuristic theory (Broue, Malle, Michel) of unipotent representations associatedto complex reflection groups. In the case of finite Coxeter group this theory isno longer heuristic: it now has a concrete meaning described in [226] in terms ofJ-rings [73].

44 G. LUSZTIG

[111] (with I.Grojnowski) A comparison ofbases of quantized enveloping algebras, 1993

At the end of 1991 there were two definitions of a canonical basis of the pluspart U+ of the quantized enveloping algebra of a Kac-Moody Lie algebra withsymmetric Cartan matrix: the algebraic one in [Kashiwara, Duke Math.J. 1991]and a topological one in [[97]. (But it was already known that, for finite types,both these definitions agree with the original definition [92], see [95],[97].) Inthis paper it is shown that these two bases agree in the general case. The newidea of this paper is a geometric interpretation of the symmetric bilinear form (, )on U+. Namely for b, b′ in the basis [97], it is shown that the rational function(b, b′) expanded in a power series in v−1 has coefficients given by the dimensionsof the equivariant Ext groups between the equivariant simple perverse sheaveswhich represent b, b′. (These Ext groups can be defined along the same linesas the equivariant homology spaces in [81].) In particular these coefficients arenatural numbers. The direct sum of the equivariant Ext groups above (for variousdegrees and various b, b′) is naturally an algebra which, by [Varagnolo and Vasserot,arxiv:0901.3992], coincides with the KLR-algebra introduced combinatorially in[Khovanov and Lauda, arxiv:0803.4121] and [Rouquier, arxiv:0812.5023].

[112] Tight monomials in quantized enveloping algebras, 1993

In this paper I show that the construction [97] in terms of quivers of the canon-ical basis of the plus part U+ of a quantized enveloping algebra can be generalizedto the case where the quiver is allowed to have loops (this was not allowed in[97]). The resulting class of algebras includes the usual U+ but also the classicalHall algebra with their canonical bases. Moreover, the plus part of a quantized(Borcherds) generalized Kac-Moody Lie algebra as described in [Kang and Schiff-mann, Adv. Math. 2006]) is in fact a subalgebra of one of our U+, and thecanonical basis described in [loc.cit.] is closely connected with the canonical basisof U+ introduced in this paper, see [Kang and Schiffmann, arxiv:0711.1948]. In the

U+ of this paper there are elements F(a)i of the canonical basis indexed by a vertex

i of the quiver and a natural number a. In the case without loops these elementsare divided powers of a single element Fi but in the general case this is not so.

In the paper I conjectured that the elements F(a)i generate the algebra U+. (This

was known from [87] in the case without loops and was proved in the paper in thecase where there is only one vertex and any number of loops.) The conjecture is

now proved by [T. Bozec, 2014]. Consider now a monomial m = F(a1)i1

F(a2)i2

...F(an)in

in the F(a)i . We say that m is tight if it belongs to the canonical basis. In this

paper I give a criterion to determine whether m is tight. The criterion is in termsof a certain positivity property of a quadratic form. Using this criterion I showthat m is always tight if there is exactly one vertex and at least two loops. I alsoinvestigate the existence of tight monomials in the loop free case of small rank. Itwas already known from [92] that in type A2 all elements of the canonical basis


are tight monomials. In the paper I show that in type A3 there is an abundance oftight monomials. In some sense (explained in the paper), 80/100 of the canonicalbasis consists of tight monomials; they fall into 8 families indexed by the variousreduced expressions of the longest Weyl group element). Later, in [N.Xi, Com-mun. in Alg. 1999], the remaining elements of the canonical basis were describedexplicitly in this case; they are not tight monomials. The tight monomials in typeA4 are described in [Y.Hu, J.Ye and X.Yue, J.Alg. 2003]. But in higher rank thereare fewer and fewer tight monomials.

[122] Quantum groups at v = ∞, 1995

The main contribution of this paper is that the idea of the J-ring (an asymptoticversion of the Hecke algebra) introduced in [73] makes sense in other contexts. Inthis paper we try to develop this idea in the case where the Hecke algebra withits canonical basis and its a-function is replaced by the modified quantum groupU with its canonical basis B introduced in [101] and an appropriate a-function on

it. This leads to a ring version at infinity U∞ of U . In the paper this is madeexplicit for quantum groups of finite type and is stated as a conjecture for affinetype. The conjecture has now been proved for type A in [K. McGerty, Int. Math.Res. Not. 2003] and in general in [J. Beck and H. Nakajima, Duke Math.J. 2004].

[126] Braid group actions and canonical bases, 1996

Let U be the quantized enveloping algebra corresponding to a given root datum.Let U+ be the plus part of U . Let Ei be the standard generators of U+. Let Ti

be the symmetries of U defined in [107, Part VI] and let B be the canonical basisof U+ defined in [107, 14.4]. In this paper I show that Ti respects B as much aspossible. More precisely, we have U+ = (U+∩T−1

i U+)⊕U+Ei and I show that the

associated projection U+ 7→ (U+ ∩ T−1i U+) applies B to a basis of U+ ∩ T−1

i U+

union with 0. Similarly we have U+ = (TiU+ ∩ U+)⊕EiU

+ and I show that theassociated projection U+ 7→ (TiU

+∩U+) applies B to a basis of TiU+∩U+ union

with 0. I then show that these bases of U+ ∩ T−1i U+, TiU

+ ∩ U+ correspondto each other under Ti. According to [Baumann, arxiv:1104.0907], an analogusresult holds when the canonical basis B is replaced by the semicanonical basis[147] assuming that the root datum is simply laced and v=1. The results of thispaper have been used in [Beck, Chari and Pressley, Duke Math.J. 1999] to give acharacterization of the canonical basis B of U+ (in the affine case) in terms of abasis B′ of U+ of PBW type, constructed using (in part) iterations of symmetriesTi; the results of this paper are used to show that any element of B′ is congruentto a unique element of B modulo v−1 times the Z[v−1]-lattice generated by B.(This extends the results of [92] in the finite type case.)

[131] Notes on unipotent classes, 1997

LetG be a semisimple almost simple algebraic group over an algebraically closed

46 G. LUSZTIG

field k whose characteristic is 0 or a good prime. In this paper I study a partitionof the unipotent variety of G into loccally closed strata, called special pieces. Eachspecial piece is a union of unipotent classes of which exactly one is special in thesense of [36] (that is the corresponding Springer representation of the Weyl groupis special in the sense of [36]; the other unipotent classes in the piece are in theclosure of the special class in the piece but not in the closure of any smaller specialpiece. The fact that this partition of G is well defined was shown by Spaltensteinin his book.

In [44] it was conjectured that any special piece is a rational homology manifold.This was later shown to be true by [Beynon and Spaltenstein, J.Alg. 1984] and[Kraft and Procesi, Asterisque 1989]. In this paper we state a refinement of thisconjecture: a special piece is the quotient of a smooth variety by a finite group.

In [44] it was also conjectured that the polynomials in q which give the numberof Fq-points of a special piece (when k = Fq) depends only on the Weyl group(not on the root system). This conjecture is proved in this paper by a complicatedcomputation. Another proof of the conjecture based on Kato’s exotic nilpotentcone was given in [Achar, Henderson and Sommers, Repres. Th., 2011]. A con-jectural explanation for why the conjecture should hold was given in [Geck andMalle, Experimental Math., 1999].

In this paper I give the following characterization of special pieces which doesn’tuse the notion of closure: two unipotent classes belong to the same special pieceif and only if the corresponding Springer representations belong to the same two-sided cell of the Weyl group.

[132] Cells in affine Weyl groups and tensor categories, 1997

The main conjecture of this paper is proved in [Bezrukavnikov, Adv.StudiesPure Math.40, Mat. Soc. Japan 2004].

[138] On quiver varieties, 1998

Theorem 5.5 has been strengthened in [Malkin, Ostrik, Vybornov, Adv.in Math.2006] where it is shown that the morphism in that Theorem is in fact an isomor-phism of algebraic varieties.

[148] Fermionic form and Betti numbers, 2000

This paper contains a conjecture which expresses the Betti numbers of theNakajima quiver varieties in terms of a certain complicated but in principle com-putable fermionic form. This conjecture has now been proved in [Kodera, Naoi,arxiv:1103.4207].

[157] Rationality properties of unipotent representations, 2002

Let G be a split connected reductive group over Fq. For each w in the Weylgroup W of G let Rw be the virtual representation of G(Fq) associated to w in [22].


Let r be a unipotent representation of G(Fq) that is, an irreducible representationappearing in Rw for some w. Let A(r) be the set of w in W such that r appearsin Rw. Let A′(r) be the set of elements of minimal length of A(r). One of themain observations of this paper is that if r is cuspidal then A′(r) is containedin a single conjugacy class C(r) of W and that for w in C(r), the multiplicityof r in Rw is equal to (−1)semisimple rank of G. From this it is deduced that aunipotent representations of G(Fq) whose character has values in rational numbersis actually defined over the rational numbers; in particular if G is of classicaltype any unipotent representation is defined over the rational numbers. (Thisis not true for unitary groups over Fq). The proof is not constructive since ituses the Hasse principle for division algebras. It is also observed that an analogueA 7→ C(A) of the correspondence r 7→ C(r) holds when r is replaced by a unipotentcuspidal character sheaf A. For example if G is of type E8/F4/G2 and A is theunique unipotent cuspidal character sheaf with unipotent support (the closureof the conjugacy class γ of a unipotent element whose centralizer has group ofcomponents S5/S4/S3) then C(A) contains an element which is ”regular” of order6/4/3 (=largest order of an element of S5/S4/S3). In the paper it is noted thatin these three cases C(A) consists of elements of a single length 40/12/4. It isinteresting that C(A) also corresponds to γ under a quite different correspondencedescribed in [197]. The rationality property of unipotent representations describedin this paper was known to me (with a different proof, also explained in the paper)since 1982 when it was the object of a lecture that I gave at a US-France Conferenceon Representation Theory in Paris. The results of this paper were presented at aconference in Rome (June 2001) and one in Isle de Berder (Bretagne) in September2001.

[167] An induction theorem for Springer’s representations, 2004

The theorem in the title was stated without proof in [48] for reductive groupsin characteristic zero and it was one of the main tools in the computation in [48]of the Springer correspondence for groups of type En. This paper (written in2001) contains a proof of that theorem, valid in arbitrary characteristic. It usesthe connection between Green functions of a reductive group over a finite field andcharacter sheaves [89] and also some arithmetic considerations.

References

[1] G. Lusztig, Model de geometrie afina plana peste un corp finit, Studii Cerc. Mat. 17

(1965)), 1337-1340.

[2] G.Lusztig, Constructia fibrarilor universale peste poliedre arbitrare, Studii Cerc. Mat. 18

(1965), 1215-1219.

[3] G.Lusztig and H.Moscovici, Demonstration du theoreme sur la suite spectrale d’un fibre

au sens de Kan, Proc. Camb. Phil. Soc. 64 (1968), 293-297.

[4] G.Lusztig, Sur les complexes elliptiques fibres, C.R. Acad. Sci. Paris(A) 266 (1968),

914-917.

48 G. LUSZTIG

[5] G.Lusztig, Sur les actions libres des groupes finis, Bull. Acad. Polon. Sci. 16 (1968),

461-463.

[6] G.Lusztig, Coomologia complexelor eliptice, Studii Cerc. Mat. 21 (1969), 38-83.

[7] G.Lusztig, A property of certain non-degenerate holomorphic vector fields, An. Univ.Timisoara 7 (1969), 73-76.

[8] G.Lusztig, J.Milnor and F.P.Peterson, Semicharacteristics and cobordism, Topology 8

(1969), 357-359.

[9] G.Lusztig, Remarks on the holomorphic Lefschetz formula, Analyse globale, Presses de

l’Univ.de Montreal, 1971, pp. 193-204.

[10] G.Lusztig and J.Dupont, On manifolds satisfying w2

1= 0, Topology 10 (1971), 81-92.

[11] G.Lusztig, Novikov’s higher signature and families of elliptic operators, J. Diff. Geom. 7(1972), 229-256.

[12] G.Lusztig, On the discrete series representations of the general linear groups over a finitefield, Bull. Amer. Math. Soc. 79 (1973), 550-554.

[13] G.Lusztig, The discrete series of GLn over a finite field, Ann. Math. Studies 81, Princeton

U.Press, 1974.

[14] G.Lusztig, Introduction to elliptic operators, Global Analysis and applications, Inter-

nat.Atomic Energy Agency, Vienna, 1974, pp. 187-193.

[15] G.Lusztig and R.W.Carter, On the modular representations of the general linear andsymmetric groups, Math. Z. 136 (1974), 193-242.

[16] G.Lusztig and R.W.Carter, Modular representations of the general linear and symmetric

groups, Proc.2nd Int.Conf. Th.Groups 1973, LNM 372, Springer Verlag, 1974, pp. 218-

220.

[17] G.Lusztig, On the discrete series representations of the classical groups over a finite field,Proc.Int.Congr.Math.,Vancouver 1974, pp. 465-470.

[18] G.Lusztig, Sur la conjecture de Macdonald, C.R. Acad. Sci. Paris(A) 280 (1975), 371-320.

[19] G.Lusztig, A note on counting nilpotent matrices of fixed rank, Bull. Lond. Math. Soc.

8 (1976), 77-80.

[20] G.Lusztig, Divisibility of projective modules of finite Chevalley groups by the Steinberg

module, Bull. Lond. Math. Soc. 8 (1976), 130-134.

[21] G.Lusztig and R.W.Carter, Modular representations of finite groups of Lie type, Proc.Lond. Math. Soc. 32 (1976), 347-384.

[22] G.Lusztig and P.Deligne, Representations of reductive groups over finite fields, Ann.Math. 103 (1976), 103-161.

[23] G.Lusztig, On the Green polynomials of classical groups, Proc. Lond. Math. Soc. 33

(1976), 443-475.

[24] G.Lusztig, Coxeter orbits and eigenspaces of Frobenius, Inv. Math. 28 (1976), 101-159.

[25] G.Lusztig, On the finiteness of the number of unipotent classes, Inv. Math. 34 (1976),

201-213.

[26] G.Lusztig, J.A.Green and G.I.Lehrer, On the degrees of certain group characters, Quart.J. Math. 27 (1976), 1-4.

[27] G.Lusztig and B.Srinivasan, The characters of the finite unitary groups, J. Alg. 49 (1977),

167-171.

[28] G.Lusztig, Classification des representations irreductibles des groupes classiques finis,

C.R. Acad. Sci. Paris(A) 284 (1977), 473-476.

[29] G.Lusztig, Irreducible representations of finite classical groups, Inv. Math. 43 (1977),125-175.

[30] G.Lusztig, Representations of finite Chevalley groups, Regional Conf. Series in Math. 39,Amer. Math. Soc., 1978.

[31] W.M.Beynon and G.Lusztig, Some numerical results on the characters of exceptional

Weyl groups, Math. Proc. Camb. Phil. Soc. 84 (1978), 417-426.


[32] G.Lusztig, Some remarks on the supercuspidal representations of p-adic semisimple

groups, Proc. Symp. Pure Math.33(1), Amer. Math. Soc., 1979, pp. 171-175.

[33] G.Lusztig, On the reflection representation of a finite Chevalley group,, Representationtheory of Lie groups, LMS Lect.Notes Ser.34, Cambridge U.Press, 1979, pp. 325-337.

[34] G.Lusztig, Unipotent representations of a finite Chevalley group of type E8, Quart. J.Math. 30 (1979), 315-338.

[35] G.Lusztig and N.Spaltenstein, Induced unipotent classes, J. Lond. Math. Soc. 19 (1979),41-52.

[36] G.Lusztig, A class of irreducible representations of a Weyl group, Proc. Kon. Nederl.

Akad.(A) 82 (1979), 323-335.

[37] D.Kazhdan and G.Lusztig, Representations of Coxeter groups and Hecke algebras, Inv.

Math. 53 (1979), 165-184.

[38] D.Kazhdan and G.Lusztig, A topological approach to Springer’s representations, Adv.

Math. 38 (1980), 222-228.

[39] D.Kazhdan and G.Lusztig, Schubert varieties and Poincare duality, Proc. Symp. Pure

Math.36, Amer. Math. Soc., 1980, pp. 185-203.

[40] G.Lusztig, Some problems in the representation theory of finite Chevalley groups, Proc.

Symp. Pure Math.37, Amer. Math. Soc., 1980, pp. 313-317.

[41] G.Lusztig, Hecke algebras and Jantzen’s generic decomposition patterns, Adv.Math. 37(1980), 121-164.

[42] G.Lusztig, On the unipotent characters of the exceptional groups over finite fields, Inv.Math. 60 (1980), 173-192.

[43] G.Lusztig, On a theorem of Benson and Curtis, J. Alg. 71 (1981), 490-498.

[44] G.Lusztig, Green polynomials and singularities of unipotent classes, Adv. Math. 42

(1981), 169-178.

[45] G.Lusztig, Unipotent characters of the symplectic and odd orthogonal groups over a finitefield, Inv. Math. 64 (1981), 263-296.

[46] G.Lusztig, Unipotent characters of the even orthogonal groups over a finite field, Trans.Amer. Math. Soc. 272 (1982), 733-751.

[47] P.Deligne and G.Lusztig, Duality for representations of a reductive group over a finite

field, J. Alg. 74 (1982), 284-291.

[48] D.Alvis and G.Lusztig, On Springer’s correspondence for simple groups of type En(n =

6, 7, 8), Math. Proc. Camb. Phil. Soc. 92 (1982), 65-72.

[49] D.Alvis and G.Lusztig, The representations and generic degrees of the Hecke algebras of

type H4, J. reine und angew. math. 336 (1982), 201-212; Erratum 449 (1994), 217-281.

[50] G.Lusztig, A class of irreducible representations of a Weyl group II, Proc. Kon. Nederl.

Akad.(A) 85 (1982), 219-226.

[51] G.Lusztig and D.Vogan, Singularities of closures of K-orbits on a flag manifold, Inv.

Math. 71 (1983), 365-379.

[52] P.Deligne and G.Lusztig, Duality for representations of a reductive group over a finitefield II, J. Alg. 81 (1983), 540-549.

[53] G.Lusztig, Singularities, character formulas and a q-analog of weight multiplicities,Asterisque 101-102 (1983), 208-229.

[54] G.Lusztig, Some examples of square integrable representations of semisimple p-adicgroups, Trans. Amer. Math. Soc. 227 (1983), 623-653.

[55] G.Lusztig, Left cells in Weyl groups, Lie groups representations, LNM 1024, Springer

Verlag, 1983, pp. 99-111.

[56] G.Lusztig, Open problems in algebraic groups, Proc.12th Int.Symp., Taniguchi Founda-

tion, Katata, 1983, pp. 14-14.

[57] G.Lusztig, Characters of reductive groups over a finite field, Ann.Math.Studies 107,

Princeton U.Press, 1984.

50 G. LUSZTIG

[58] G.Lusztig, Characters of reductive groups over finite fields, Proc.Int.Congr.Math. Warsaw

1983, North Holland, 1984, pp. 877-880.

[59] G.Lusztig, Intersection cohomology complexes on a reductive group, Inv. Math. 75 (1984),205-272.

[60] G.Lusztig, Cells in affine Weyl groups, Algebraic groups and related topics, Adv. Stud.Pure Math. 6, North-Holland and Kinokuniya, 1985, pp. 255-287.

[61] G.Lusztig and N.Spaltenstein, On the generalized Springer correspondence for classical

groups, Algebraic groups and related topics, Adv. Stud. Pure Math. 6, North-Holland

and Kinokuniya, 1985, pp. 289-316.

[62] G.Lusztig, The two sided cells of the affine Weyl group of type A, Infinite dimensionalgroups with applications, MSRI Publ.4, Springer Verlag, 1985, pp. 275-283.

[63] G.Lusztig, Character sheaves I, Adv. Math. 56 (1985), 193-237.

[64] G.Lusztig, Character sheaves II, Adv.Math. 57 (1985), 226-265.

[65] G.Lusztig, Character sheaves III, Adv.Math. 57 (1985), 266-315.

[66] G.Lusztig, Equivariant K-theory and representations of Hecke algebras, Proc. Amer.

Math. Soc. 94 (1985), 337-342.

[67] D.Kazhdan and G.Lusztig, Equivariant K-theory and representations of Hecke algebras

II, Inv. Math. 80 (1985), 209-231.

[68] G.Lusztig, Character sheaves IV, Adv. Math. 59 (1986), 1-63.

[69] G.Lusztig, Character sheaves V, Adv. Math. 61 (1986), 103-155.

[70] G.Lusztig, Sur les cellules gauches des groupes de Weyl, C.R. Acad. Sci. Paris(A) 302

(1986), 5-8.

[71] G.Lusztig, On the character values of finite Chevalley groups at unipotent elements, J.Alg. 104 (1986), 146-194.

[72] D.Kazhdan and G.Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras,Inv. Math. 87 (1987), 153-215.

[73] G.Lusztig, Cells in affine Weyl groups II, J. Alg. 109 (1987), 536-548.

[74] G.Lusztig, Fourier transforms on a semisimple Lie algebra over Fq,, Algebraic GroupsUtrecht 1986, LNM 1271, Springer Verlag, 1987, pp. 177-188.

[75] G.Lusztig, Cells in affine Weyl groups III, J. Fac. Sci. Tokyo U.(IA) 34 (1987), 223-243.

[76] G.Lusztig, Introduction to character sheaves, Proc. Symp. Pure Math. 47(1), Amer. Math.Soc., 1987, pp. 165-180.

[77] G.Lusztig, Leading coefficients of character values of Hecke algebras, Proc. Symp. Pure

Math. 47(2), Amer. Math. Soc., 1987, pp. 235-262.

[78] C.De Concini, G.Lusztig and C.Procesi, Homology of the zero set of a nilpotent vector

field on a flag manifold, J. Amer. Math. Soc. 1 (1988), 15-34.

[79] G.Lusztig, Quantum deformations of certain simple modules over enveloping algebras,

Adv.Math. 70 (1988), 237-249.

[80] D.Kazhdan and G.Lusztig, Fixed point varieties on affine flag manifolds, Isr. J. Math.62 (1988), 129-168.

[81] G.Lusztig, Cuspidal local systems and graded Hecke algebras I, Publ. Math. I.H.E.S. 67(1988), 145-202.

[82] G.Lusztig and N.Xi, Canonical left cells in affine Weyl groups, Adv.Math. 72 (1988),

284-288.

[83] G.Lusztig, On representations of reductive groups with disconnected center, Asterisque

168 (1988), 157-166.

[84] G.Lusztig, Modular representations and quantum groups, Contemp. Math. 82 (1989),

59-77.

[85] G.Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989),599-635.

[86] G.Lusztig, Cells in affine Weyl groups IV, J. Fac. Sci. Tokyo U.(IA) 36 (1989), 297-328.

[87] G.Lusztig, Representations of affine Hecke algebras, Asterisque 171-172 (1989), 73-84.


[88] G.Lusztig, On quantum groups, J. Alg. 131 (1990), 466-475.

[89] G.Lusztig, Green functions and character sheaves, Ann. Math. 131 (1990), 355-408.

[90] G.Lusztig, Finite dimensional Hopf algebras arising from quantized universal envelopingalgebras, J. Amer. Math. Soc. 3 (1990), 257-296.

[91] G.Lusztig, Quantum groups at roots of 1, Geom.Ded. 35 (1990), 89-114.

[92] G.Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math.

Soc. 3 (1990), 447-498.

[93] A.A.Beilinson, G.Lusztig and R.MacPherson, A geometric setting for the quantum de-formation of GLn, Duke Math. J. 61 (1990), 655-677.

[94] G.Lusztig, Symmetric spaces over a finite field, The Grothendieck Festschrift III, Progr.

in Math. 88, Birkhauser Boston, 1990, pp. 57-81.

[95] G.Lusztig, Canonical bases arising from quantized enveloping algebras II, Progr.of Theor.Phys. Suppl. 102 (1990), 175-201.

[96] D.Kazhdan and G.Lusztig, Affine Lie algebras and quantum groups, Int. Math. Res.

Notices (1991), 21-29.

[97] G.Lusztig, Quivers, perverse sheaves and enveloping algebras J. Amer. Math. Soc. 4

(1991), 365-421.

[98] G.Lusztig and J.M.Smelt, Fixed point varieties in the space of lattices Bull. Lond. Math.

Soc. 23 (1991), 213-218.

[99] G.Lusztig, Intersection cohomology methods in representation theory, Proc. Int. Congr.

Math. Kyoto 1990, Springer Verlag, 1991, pp. 155-174.

[100] G.Lusztig, A unipotent support for irreducible representations, Adv. Math. 94 (1992),

139-179.

[101] G.Lusztig, Canonical bases in tensor products, Proc. Nat. Acad. Sci. 89 (1992), 8177-

8179.

[102] G.Lusztig, Remarks on computing irreducible characters, J. Amer. Math. Soc. 5 (1992),971-986.

[103] G.Lusztig, Introduction to quantized enveloping algebras, Progr.in Math.105, Birkhauser

Boston, 1992, pp. 49-65.

[104] G.Lusztig, Affine quivers and canonical bases, Publ. Math. I.H.E.S. 76 (1992), 111-163.

[105] G.Lusztig and J.Tits, The inverse of a Cartan matrix, An.Univ.Timisoara 30 (1992),

17-23.

[106] I.Grojnowski and G.Lusztig, On bases of irreducible representations of quantum GLn,

Kazhdan-Lusztig theory and related topics, Contemp.Math.139, 1992, pp. 167-174.

[107] G.Lusztig, Introduction to quantum groups, Progr.in Math.110, Birkhauser Boston, 1993.

[108] D.Kazhdan and G.Lusztig, Tensor structures arising from affine Lie algebras I, J. Amer.

Math. Soc. 6 (1993), 905-947.

[109] D.Kazhdan and G.Lusztig, Tensor structures arising from affine Lie algebras II, J. Amer.

Math. Soc. 6 (1993), 949-1011.

[110] G.Lusztig, Coxeter groups and unipotent representations, Asterisque 212 (1993), 191-203.

[111] I.Grojnowski and G.Lusztig, A comparison of bases of quantized enveloping algebras,

Linear algebraic groups and their representations, Contemp.Math.153, 1993, pp. 11-19.

[112] G.Lusztig, Tight monomials in quantized enveloping algebras, Quantum deformations ofalgebras and their representations, ed. A.Joseph et al., Isr. Math. Conf. Proc. 7, Amer.

Math. Soc., 1993, pp. 117-132.

[113] G.Lusztig, Exotic Fourier transform, Duke Math.J. 73 (1994), 227-241.

[114] G.Lusztig, Vanishing properties of cuspidal local systems, Proc. Nat. Acad. Sci. 91 (1994),1438-1439.

[115] D.Kazhdan and G.Lusztig, Tensor structures arising from affine Lie algebras III, J.

Amer. Math. Soc. 7 (1994), 335-381.

52 G. LUSZTIG

[116] D.Kazhdan and G.Lusztig, Tensor structures arising from affine Lie algebras IV, J.

Amer. Math. Soc. 7 (1994), 383-453.

[117] G.Lusztig, Monodromic systems on affine flag manifolds, Proc. Roy. Soc. Lond.(A) 445

(1994), 231-246; Errata 450 (1995), 731-732.

[118] G.Lusztig, Problems on canonical bases, Algebraic groups and their generalizations:quantum and infinite dimensionalmethods, Proc. Symp. Pure Math. 56(2), Amer. Math.

Soc., 1994, pp. 169-176.

[119] G.Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr.in Math.123, Birkhauser Boston, 1994, pp. 531-568.

[120] G.Lusztig, Study of perverse sheaves arising from graded Lie algebras, Adv.Math. 112

(1995), 147-217.

[121] G.Lusztig, Cuspidal local systems and graded Hecke algebras II, Representations of groups,

ed. B.Allison et al., Canad. Math. Soc. Conf. Proc.16, Amer. Math. Soc., 1995, pp. 217-

275.

[122] G.Lusztig, Quantum groups at v = ∞, Functional analysis on the eve of the 21st century,

vol.I, Progr.in Math. 131, Birkhauser Boston, 1995, pp. 199-221.

[123] G.Lusztig, Classification of unipotent representations of simple p-adic groups, Int. Math.Res. Notices (1995), 517-589.

[124] G.Lusztig, An algebraic-geometric parametrization of the canonical basis, Adv. Math.

120 (1996), 173-190.

[125] G.Lusztig, Affine Weyl groups and conjugacy classes in Weyl groups, Transform. Groups(1996), 83-97.

[126] G.Lusztig, Braid group actions and canonical bases, Adv. Math. 122 (1996), 237-261.

[127] G.Lusztig, Non local finiteness of a W -graph, Represent.Th 1 (1997), 25-30.

[128] G.Lusztig, Cohomology of classifying spaces and hermitian representations, Represent.Th.

1 (1997), 31-36.

[129] C.K.Fan and G.Lusztig, Factorization of certain exponentials in Lie groups, Algebraicgroups and Lie groups, ed. G.I.Lehrer, Cambridge U.Press, 1997, pp. 215-218.

[130] G.Lusztig, Total positivity and canonical bases, Algebraic groups and Lie groups, ed.

G.I.Lehrer, Cambridge U.Press, 1997, pp. 281-295.

[131] G.Lusztig, Notes on unipotent classes, Asian J.Math. 1 (1997), 194-207.

[132] G.Lusztig, Cells in affine Weyl groups and tensor categories, Adv. Math. 129 (1997),

85-98.

[133] G.Lusztig, Periodic W -graphs, Represent.Th. 1 (1997), 207-279.

[134] G.Lusztig, A comparison of two graphs, Int. Math. Res. Notices (1997), 639-640.

[135] G.Lusztig, Constructible functions on the Steinberg variety, Adv. Math. 130 (1997),

287-310.

[136] G.Lusztig, Total positivity in partial flag manifolds, Represent.Th. 2 (1998), 70-78.

[137] G.Lusztig, Introduction to total positivity, Positivity in Lie theory: open problems, ed.

J.Hilgert et al., de Gruyter, 1998, pp. 133-145.

[138] G.Lusztig, On quiver varieties, Adv.Math. 136 (1998), 141-182.

[139] G.Lusztig, Canonical bases and Hall algebras, Representation Theories and AlgebraicGeometry, ed. A.Broer et al., Kluwer Acad.Publ., 1998, pp. 365-399.

[140] G.Lusztig, Bases in equivariant K-theory, Represent.Th. 2 (1998), 298-369.

[141] G.Lusztig, Homology bases arising from reductive groups over a finite field, Algebraicgroups and their representations, ed. R.W.Carter et al., Kluwer Acad. Publ., 1998, pp. 53-

72.

[142] G.Lusztig, Aperiodicity in quantum affine gln, Asian J. Math. 3 (1999), 147-178.

[143] G.Lusztig, Bases in equivariant K-theory II, Represent. Th. 3 (1999), 281-353.

[144] G.Lusztig, A survey of group representations, Nieuw Archief voor Wiskunde 17 (1999),

483-489.


[145] G.Lusztig, Subregular nilpotent elements and bases in K-theory, Canad. J. Math. 51

(1999), 1194-1225.

[146] G.Lusztig, Recollections about my teacher, Michael Atiyah, Asian J. Math. 3 (1999), iv-v.

[147] G.Lusztig, Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000),

129-139.

[148] G.Lusztig, Fermionic form and Betti numbers, arxiv:QA/0005010.

[149] G.Lusztig, Quiver varieties and Weyl group actions, Ann. Inst. Fourier 50 (2000), 461-

489.

[150] G.Lusztig, G(Fq)-invariants in irreducible G(Fq2)-modules, Represent. Th. 4 (2000),

446-465.

[151] G.Lusztig, Remarks on quiver varieties, Duke Math. J. 105 (2000), 239-265.

[152] G.Lusztig, Transfer maps for quantum affine sln, Representations and quantizations, ed.J.Wang et al., China Higher Ed.Press and Springer Verlag, 2000.

[153] G.Lusztig, Representation theory in characteristic p, Taniguchi Conf. on Math. Nara’98,

Adv. Stud. Pure Math. 31, Math. Soc. Japan, 2001, pp. 167-178.

[154] G.Lusztig, Cuspidal local systems and graded Hecke algebras III, Represent.Th. 6 (2002),

202-242.

[155] G.Lusztig, Classification of unipotent representations of simple p-adic groups II, Repre-

sent.Th. 6 (2002), 243-289.

[156] G.Lusztig, Constructible functions on varieties attached to quivers, Studies in memoryof I. Schur, Progr. in Math. 210, Birkhauser Boston, 2002, pp. 177-223.

[157] G.Lusztig, Rationality properties of unipotent representations, J. Alg. 258 (2002), 1-22.

[158] G.Lusztig, Notes on affine Hecke algebras, Iwahori-Hecke algebras and their representa-tion theory, ed. M.W.Baldoni et al., LNM 1804, Springer Verlag, 2002, pp. 71-103.

[159] G.Lusztig, Hecke algebras with unequal parameters, CRM Monograph Ser.18, Amer.Math. Soc. 2003,136p, additional material in version 2 (2014), arxiv:math/0208154.

[160] G.Lusztig, Homomorphisms of the alternating group A5 into reductive groups, J. Alg.

260 (2003), 298-322.

[161] G.Lusztig, Character sheaves on disconnected groups I, Represent. Th. 7 (2003), 374-403;

Errata 8 (2004), 179-179.

[162] G.Lusztig, Representations of reductive groups over finite rings, Represent. Th. 8 (2004),

1-14.

[163] G.Lusztig, Character sheaves on disconnected groups II, Represent. Th. 8 (2004), 72-124.

[164] G.Lusztig, Character sheaves on disconnected groups III, Represent. Th. 8 (2004), 125-

144.

[165] G.Lusztig, Character sheaves on disconnected groups IV, Represent. Th. 8 (2004), 145-178.

[166] G.Lusztig, Parabolic character sheaves I, Moscow Math.J. 4 (2004), 153-179.

[167] G.Lusztig, An induction theorem for Springer’s representations, Adv.Stud.Pure Math.40,Math. Soc. Japan, Kinokuniya, 2004, pp. 253-259.

[168] G.Lusztig, Character sheaves on disconnected groups V, Represent. Th. 8 (2004), 346-376.

[169] G.Lusztig, Character sheaves on disconnected groups VI, Represent.Th. 8 (2004), 377-

413.

[170] G.Lusztig, Parabolic character sheaves II, Moscow Math.J. 4 (2004), 869-896.

[171] G.Lusztig, Convolution of almost characters, Asian J. Math. 8 (2004), 769-772.

[172] G.Lusztig, Character sheaves on disconnected groups VII, Represent. Th. 9 (2005), 209-266.

[173] G.Lusztig, Unipotent elements in small characteristic, Transform. Groups 10 (2005),

449-487.

[174] G.Lusztig, Character sheaves and generalizations, The Unity of Mathematics,

ed. P.Etingof et al., Progress in Math.244, Birkhauser Boston, 2006, pp. 443-455.

54 G. LUSZTIG

[175] G.Lusztig, A q-analogue of an identity of N.Wallach, Studies in Lie theory,

ed. J.Bernstein et al., Progress in Math. 243, Birkhauser Boston, 2006, pp. 405-410.

[176] G.Lusztig, Character sheaves on disconnected groups VIII, Represent. Th. 10 (2006),314-352.

[177] G.Lusztig, Character sheaves on disconnected groups IX, Represent. Th. 10 (2006), 353-379.

[178] G.Lusztig, A class of perverse sheaves on a partial flag manifold, Represent. Th. 11

(2007), 122-171.

[179] X.He and G.Lusztig, Singular supports for character sheaves on a group compactification,

Geom. and Funct.Analysis 17 (2007), 1915-1923.

[180] G.Lusztig, Irreducible representations of finite spin groups, Represent. Th. 12 (2008),

1-36.

[181] G.Lusztig, A survey of total positivity, Milan J. Math. 76 (2007), 1-10.

[182] G.Lusztig, Generic character sheaves on disconnected groups and character values, Rep-

resent. Th. 12 (2008), 225-235.

[183] G.Lusztig, Unipotent elements in small characteristic II, Transform. Groups 13 (2008),773-797.

[184] G.Lusztig, Study of a Z-form of the coordinate ring of a reductive group, J. Amer. Math.Soc. 22 (2009), 739-769.

[185] S.Kumar, G.Lusztig and D.Prasad, Characters of simplylaced nonconnected groups ver-

sus characters of nonsimplylaced connected groups, Representation theory, ed. Z.Lin,

Contemp. Math. 478, 2009, pp. 99-101.

[186] G.Lusztig, Twelve bridges from a reductive group to its Langlands dual, Representationtheory, ed. Z.Lin, Contemp. Math.478, 2009, pp. 125-143.

[187] G.Lusztig, Character sheaves on disconnected groups X, Represent. Th. 13 (2009), 82-140.

[188] G.Lusztig, Unipotent classes and special Weyl group representations, J. Alg. 321 (2009),

3418-3449.

[189] G.Lusztig, Remarks on Springer’s representations, Represent. Th. 13 (2009), 391-400.

[190] G.Lusztig, Notes on character sheaves, Moscow Math.J. 9 (2009), 91-109.

[191] G.Lusztig, Graded Lie algebras and intersection cohomology, Representation theory of al-gebraic groups and quantum groups, ed. A.Gyoja et al., Progress in Math.284, Birkhauser,

2010, pp. 191-224.

[192] G.Lusztig, Unipotent elements in small characteristic IV, Transform. Groups 14 (2010).

[193] G.Lusztig, Parabolic character sheaves III, Moscow Math.J. 10 (2010), 603-609.

[194] G.Lusztig, Unipotent elements in small characteristic III, J. Alg. 329 (2011), 163-189.

[195] G.Lusztig, Piecewise linear parametrization of canonical bases, Pure Appl. Math. Quart.

7 (2011), 783-796.

[196] G.Lusztig, On some partitions of a flag manifold, Asian J. Math. 15 (2011), 1-8.

[197] G.Lusztig, From conjugacy classes in the Weyl group to unipotent classes, Represent.Th.

15 (2011), 494-530.

[198] G.Lusztig, From groups to symmetric spaces, Contemp. Math. 557 (2011), 245-258.

[199] G.Lusztig, Study of antiorbital complexes, Contemp. Math. 557 (2011), 259-287.

[ 200] G.Lusztig, On C-small conjugacy classes in a reductive group, Transfor. Groups 16

(2011), 807-825.

[201] G.Lusztig, Bruhat decomposition and applications, arxiv:1006.5004.

[202] G.Lusztig, On certain varieties attached to a Weyl group element, Bull. Inst. Math. Acad.Sinica (N.S.) 6 (2011), 377-414.

[203] X.He and G.Lusztig, A generalization of Steinberg’s cross-section, J. Amer. Math. Soc.

25 (2012), 739-757.

[204] G.Lusztig, Elliptic elements in a Weyl group: a homogeneity property, Represent. Th.

16 (2012), 127-151.


[205] G.Lusztig, From conjugacy classes in the Weyl group to unipotent classes II, Represent.

Th. 16 (2012), 189-211.

[206] G.Lusztig, On the cleanness of cuspidal character sheaves, Moscow Math.J. 12 (2012),621-631.

[207 ] G.Lusztig and T.Xue, Elliptic Weyl group elements and unipotent isometries with p = 2,

Represent. Th. 16 (2012), 270-275.

[208] G.Lusztig and D.Vogan, Hecke algebras and involutions in Weyl groups, Bull. Inst. Math.

Acad. Sinica(N.S.) 7 (2012), 323-354.

[209] G.Lusztig, A bar operator for involutions in a Coxeter group, Bull. Inst. Math. Acad.Sinica (N.S.) 7 (2012), 355-404.

[210] G.Lusztig, From conjugacy classes in the Weyl group to unipotent classes III, Represent.

Th. 16 (2012), 450-488.

[211] G.Lusztig, On the representations of disconnected reductive groups over Fq, ”Recent

developments in Lie Algebras, Groups and Representation theory, ed. K.Misra, Proc.Symp. Pure Math., vol. 86, Amer. Math. Soc., 2012.

[212] G.Lusztig and Z.Yun, A (-q)-analogue of weight multiplicities, Jour. Ramanujan Math.

Soc. 29A (2013), 311-340.

[213] J.-L.Kim and G.Lusztig, On the characters of unipotent representations of a semisimple

p-adic group, Represent. Th. 17 (2013), 426-441.

[214] G.Lusztig, Asymptotic Hecke algebras and involutions, Perspectives in RepresentationTheory, ed. P.Etingof et.al., Contemp.Math.610, 2014, pp. 267-278.

[215] G.Lusztig, Families and Springer’s correspondence, Pacific J.Math. 267 (2014), 431-450.

[216] G.Lusztig, Restriction of a character sheaf to conjugacy classes, Bulletin Mathem. 58(2015), 297-309.

[217] J.-L.Kim and G.Lusztig, On the Steinberg character of a semisimple p-adic group, Pacific

J.Math. 265 (2013), 499-509.

[218] G.Lusztig and D.Vogan, Quasisplit Hecke algebras and symmetric spaces, Duke Math. J.

163 (2014), 983-1070.

[219] G.Lusztig, Unipotent almost characters of simple p-adic groups, Asterisque 369-370

(2015), 243-267.

[220] G.Lusztig, Unipotent almost characters of simple p-adic groups, II, Transfor. Gr. 19

(2014), 527-547.

[221] G.Lusztig, Distinguished conjugacy classes and elliptic Weyl group elements, Repre-

sent.Th. 18 (2014), 223-277.

[222] G.Lusztig, On conjugacy classes in a reductive group, Representations of ReductiveGroups, Progr.in Math. 312, Birkhauser, 2015, pp. 333-363.

[223] G.Lusztig, Truncated convolution of character sheaves, Bull. Inst. Math. Acad. Sinica

(N.S.) 10 (2015), 1-72.

[224] G.Lusztig, On conjugacy classes in the Lie group E8, arxiv:1309.1382.

[225] G.Lusztig and D.Vogan, Hecke algebras and involutions in Coxeter groups, Representa-

tions of Reductive Groups, Progr.in Math. 312, Birkhauser, 2015, pp. 365-398.

[226] G.Lusztig, Unipotent representations as a categorical centre, Represent.Th. 19 (2015),

211-235.

[227] G.Lusztig, Exceptional representations of Weyl groups, J. Alg. 475 (2017), 14-20.

[228] G.Lusztig, Action of longest element on a Hecke algebra cell module, Pacific.J.Math. 279(2015), 383-396.

[229] G.Lusztig, On the character of certain irreducible modular representations, Represent.

Th. 19 (2015), 3-8.

[230] G.Lusztig, Algebraic and geometric methods in representation theory, arxiv:1409.8003.

[231] G.Lusztig, Some power series involving involutions in Coxeter groups, Repres.Th. 19

(2015), 281-289.

56 G. LUSZTIG

[232] G.Lusztig, Nonsplit Hecke algebras and perverse sheaves, Selecta Math. 22 (2016), 1953-

1986.[233] G.Lusztig and G.Williamson, On the character of certain tilting modules, submitted.

[234] G.Lusztig, Non-unipotent character sheaves as a categorical centre, Bull. Inst. Math.Acad. Sinica (N.S.) 11 (2016), 603-731.

[235] G.Lusztig, An involution based left ideal in the Hecke algebra, Represent .Th. 20 (2016),

172-186.[236] G.Lusztig, Generic character sheaves on groups over k[ǫ]/(ǫr), Contemp. Math. 683

(2017), 227-246.

[237] G.Lusztig, Generalized Springer theory and weight functions, Ann. Univ. Ferrara Sez.VIISci. Mat. 63 (2017), 159-167.

[238] G.Lusztig, On the definition of almost characters (to appear).[239] G.Lusztig, Special representation of Weyl groups: a positivity property, Adv.Math. (to

appear).

[240] G.Lusztig and Z.Yun, Z/m-graded Lie algebras and perverse sheaves I, Represent.Th. 21(2017), 277-321.

[241] G.Lusztig, The canonical basis of the quantum adjoint representation, J. Comb. Alg. 1

(2017), 45-57.[242] G.Lusztig and Z.Yun, Z/m-graded Lie algebras and perverse sheaves II, Represent.Th.

21 (2007), 322-353 (to appear).[243] G.Lusztig and Z.Yun, Z/m-graded Lie algebras and perverse sheaves III: graded double

affine Hecke algebra, arxiv:1602.05244, submitted.

[244] G.Lusztig, On the generalized Springer correspondence, arxiv:1608:02222.[245] G.Lusztig, Non-unipotent representations and categorical centers, Bull. Inst. Math. Acad.

Sinica (N.S.) 12 (2017), 205-296.

[246] G.Lusztig and G,Williamson, Billiards and tilting characters of SL3 jour arxiv:1703.05898,submitted.

[247] G.Lusztig, Conjugacy classes in reductive groups and two-sided cells, arxiv:1706.02389.[***] G.Lusztig, Comments on my papers, arxiv:1707.09368.

[248] G.Lusztig, Lifting involutions in a Weyl group to the torus normalizer, arxiv:1709:08589.

[249] G.Lusztig, Hecke modules based on involutions in extended Weyl groups, arxiv:1710.03670.

Department of Mathematics, MIT, Cambridge MA 02139

arxiv:1707.09368v2 [math.rt] 11 oct 2017 · pdf filemoscovici, gromov, higson and ... and...

Documents